What is the term in spiritual theology to denote the aspect that increases within a believer as God carries his work to completion (Phil 1:6)?

Ongoing sanctification is the simplest, clearest term I can think of. Sanctification is explained in detail in the New Testament, and it indicates something received by the person who puts faith in Christ Jesus. This is stated here:

"So then, just as you received Christ Jesus as Lord, continue to live in him, rooted and built up in him, strengthened in the faith as you were taught, and overflowing with thankfulness." Colossians 2:6 N.I.V. 1987 ed.

To 'receive' Christ Jesus is to receive sanctification, for no unsanctified person can receive Christ.

Having received this sanctification, it is clear that the person must continue living 'in Christ'. It is not a gift to be received then put to one side, as it were, as if that was the end of the matter. No. Sanctification can be seen being worked out in the believer for the rest of their life until that glorious day we confidently look forward to when:

"...he who began a good work in you will carry it on to completion until the day of Christ Jesus." Philippians 1:6

A few verses further on, Paul adds that such Christians are to:

"...continue to work out your salvation with fear and trembling, for it is God who works in you to will and to act according to his good purpose." (2:13)

There is the on-going aspect of a salvation already received. It is at work in the believer who fears the Lord, aware of the awesome privilege of having God work in them, for his glory (not for their salvation, because they have already been given salvation.) That is why Paul gives as a reason to eagerly await for our Lord Jesus Christ to be revealed,

"He will keep you strong to the end, so that you will be blameless on the day of our Lord Jesus Christ. God, who has called you into fellowship with his Son Jesus Christ our Lord, is faithful." 1 Corinthians 1:8-9

It is because God is righteous and faithful, and has justified and sanctified the believer on the basis of his righteousness, that we can rejoice in the changed status of those who God brings into that intimate fellowship with his Son. That may be be why the writer to Hebrew Christians spoke of "things that accompany salvation" (6:9). Salvation is not a "thing", like an object. It is a changed relationship, and once that change happens, there are accompanying matters (or 'things') that evidence now having salvation.

I hope this clears a few of the points raised. But, in view of your edit to your question, and your request that I enlarge in view of that, here are some additional points.

Receiving the Holy Spirit is consequent upon the experience of justification, which is the reason for sanctification, and which is the basis of 'ongoing sanctification'. I appreciate that is not a philosophical term, but it does deal with a spiritual theology that denotes this 'increase' in a believer. Yet more can be said.

You speak of "a wall that separates theology from philosophy/ psychology in the area of spirituality" and of "the ivory tower" of theology (which Calvinists are inclined to 'inhabit'?) Let me respond with a philosophical poem about how God reaches down from heaven to teach a searching individual the importance of Truth. It was written in the 14th century (so, no Calvinists involved) by William Langland (translated here by Ronald Tamplin) and is one of the great works of English literature: Piers Plowman - Love in Action - Passus 1

"Truth proclaims it: love the wonder healer -

No blemish left, if that herb is used.

As God wished, the world was shaped in love.

Revealed to Moses, it was the best of things,

Heaven's image, priceless, the plant of peace.

But Heaven could not contain the weight of love,

Till, here on earth, it fed to the full, took

Flesh and blood. And after that, no leaf there was

On tree so light as love, mobile in air, plunging

As a needle point, no steel could stop it,

Nor castle wall. So, on earth as in heaven,

Love leads God's people, like a mayor,

Agent between the commons and the king. Love

Directs all, frames law, fixes fines

For the people's crimes. Know it for sure,

Love come surging from the power of God,

Its source, its mountain spring, the human heart."

"Speaking the truth in love" as per Ephesians 4:15 is the spiritual, theological term.

"Ongoing sanctification" works. But I think the term the OP is really looking for is theosis. Originally this comes from the tradition of the Greek church, but it was also expressed by Reformers such as Jonathan Edwards and possibly John Calvin. Recently, Reformed theologians have given increasing attention to the concept. Here is a summary of the ideas from the evangelical publication Themelios:

Orthodox writers use this Greek word to refer both to humanity’s initial vocation (the task which God gave to Adam and Eve at creation) and to salvation. The word theosis is translated ‘deification’ in English and is thus very problematic for most Western evangelicals. However, we should recognise immediately that deification does not imply that people actually become gods in any ontological sense at all; the Orthodox affirm that God is unique and transcendent, just as evangelicals do. Rather, by theosis the Orthodox mean the process of acquiring godly characteristics, gaining immortality and incorruptibility, and experiencing communion with God. As a result, deification corresponds somewhat to concepts which evangelicals describe using the terms sanctification, eternal life, and fellowship or relationship with God.

• A discussion of theosis in Calvin's thought can be found here: The question of deification in the theology of John Calvin

• Edwards' thought is examined in the Harvard Theological Review's article Jonathan Edwards's Reformed Doctrine of Theosis

• Contemporary responses to the concept in Reformed theology can be found in Reformed Theosis?: A Response to Gannon Murphy by Myk Habets.

The epistle to the saints and faithful in Ephesus soars higher into the 'heavenlies' (Ephesians 1:20, literal) than perhaps any other. And the church of Ephesus, addressed by the Son of man, Revelation 2:1, is commended for works, labour, patience, withstanding evil, trying false apostles, forbearance and not fainting.

These were mature Christians, tried and tested, very experienced, separate from the world, faithful followers.

Nevertheless, this they lacked, and that lack so significant that, did they not repent of that lack, they would no longer be a church, and would, of necessity, need to leave Ephesus and join with other congregations, as individuals, due to their corporate failure.

'Thou hast left thy first love'.

Thus says the Son of man to the Ephesian saints in Revelation 2:4.

For what is all worth, without love ?

'Though I speak with the tongue of men or angels ; though I have prophecy ; and understand all mysteries and all knowledge ; and though I have all faith ... and have not charity - I am nothing. '

Thus says Paul the apostle in 1 Corinthians 13:1-3.

Faith, itself, 'works by love', Galatians 5:6, else, it is not faith but intellectual acumen. Heady and high-minded, pompous and puffed up.

And this is God's commandment, 1 John 3:23, 'That we should believe on the name of his Son, Jesus Christ, and love one another'.

So the love prompts the faith and the faith results in love.

Such that Paul prays that 'the Lord make you to increase and abound in love one to another and toward all even as we to you', 1 Thessalonians 3:12 (KJV).

'Increase and abound in love' is certainly, therefore, an 'aspect that increases within a believer as God carries his work to completion' and is an aspect which, if lacking, is a matter so serious that a particular church can lose its status as a church should that church depart from it.

Nor is this to be mistaken with natural affection or the kind of love that nature has for its own, for the love of which the New Testament speaks is 'the love of God' that 'is shed abroad in our hearts by the Holy Spirit who is given to us', Romans 5:5.

Divine love, shed abroad, in the heart by that Person given, in an anointing, by God himself. And that same love shown one to another, increasing and abounding.

The term that, most conspicuously, stands out as something that increases within a believer as God carries his work to completion : is 'love'.

All references are to the KJV.

Many waters cannot quench love, neither can the floods drown it: if a man would give all the substance of his house for love, it would utterly be contemned. Song of Solomon 8:7 KJV.

Who shall separate us from the love of Christ ? Romans 8:35 KJV.