How does leaving a 'gift at the altar' reconcile with reformed views of the need to 'approach the throne of grace' in everything?

Holy Spirit detailed through the golden mouth of John Chrysostom how to understand Matthew 5:23-24 in HOMILIES ON MATTHEW, chapter XVI, paragraph 12.

"O goodness! O exceeding love to man! He makes no account of the honor due unto Himself, for the sake of our love towards our neighbor; implying that not at all from any enmity, nor out of any desire to punish, had He uttered those former threatenings, but out of very tender affection. For what can be milder than these sayings? "Let my service," saith he, "be interrupted, that thy love may continue; since this also is a sacrifice, thy being reconciled to thy brother." Yea, for this cause He said not, "after the offering," or "before the offering;" but, while the very gift lies there, and when the sacrifice is already beginning, He sends thee to be reconciled to thy brother; and neither after removing that which lies before us, nor before presenting the gift, but while it lies in the midst, He bids thee hasten thither...."

Regarding

Is there any sin that we must straighten out before approaching the throne of grace?

I think that "approaching the throne of grace" must have an essence, a content, and God stated the most important one: the love. Physical presence with a gift brought to the altar is an exterior form of the sacrifice, just a symbol for the essence of the sacrifice: changing of heart's state.

This gift at the altar is literally to be understood as the Fellowship or Peace offering (Leviticus chapters 3 and 7) and figuratively as that close personal relationship that we enjoy with God the Father when we are walking in the light with no thing troubling our conscience.

and indeed our fellowship is with the Father and with his Son Jesus Christ. And we are writing these things so that our joy may be complete. This is the message we have heard from him and proclaim to you, that God is light, and in him is no darkness at all. If we say we have fellowship with him while we walk in darkness, we lie and do not practice the truth. But if we walk in the light, as he is in the light, we have fellowship with one another (we and the Father), and the blood of Jesus his Son cleanses us from all sin.  If we say we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sins, he is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. If we say we have not sinned, we make him a liar, and his word is not in us. - 1 John 1:3b-10

This is well in keeping, then, with what John continuously points to as the primary signifier of our fellowship with God, our love for each other.

Beloved, if our heart does not condemn us, we have confidence before God; and whatever we ask we receive from him, because we keep his commandments and do what pleases him. And this is his commandment, that we believe in the name of his Son Jesus Christ and love one another, just as he has commanded us. Whoever keeps his commandments abides in God, and God in him. And by this we know that he abides in us, by the Spirit whom he has given us. - 1 John 3:21-24

The alter represents our relationship with God. For under the Old Covenant that relationship was maintained through the temple and the Levitical priesthood, sacrifices, offerings, etc.  Under the New Covenant we (born again believers) are the temple both corporately and individually, and we are the priests. There is no more any altar for Christ has made one offering for all and we can add nothing to it nor can it be repeated or undone.

The throne of grace is that place where we may find mercy and grace from God through Christ in time of need. When our fellowship with God the Father is interrupted by loveless living mercifully brought to our consciences by the Holy Spirit we must confess our sin and plead for mercy and grace at the throne. There we will find that "he is faithful and just to forgive us our sins and to cleanse us from all unrighteousness." In this way our fellowship with Father is restored and our 'gifts' may be once more placed on the 'altar'.

If any thought or word or action of a child of God were able to sever the Father/Child relationship, then the throne of grace would be as unavailable as it was prior to faith. The only way to the throne of grace is through the Cross and that access, once received, is never revoked. The passage in question has to do with our relationship as child to Father not of child to Father.

When Jesus was talking about leaving a gift at the altar, He was speaking to people who understood that He was speaking literally. There was a physical altar in a physical place to which these people brought literal sacrifices commanded by God.

What Jesus teaches here is that the Torah descended from higher principles, which established priorities that were higher than those of offering animals up on the altar.

Be very diligent to understand that reconciling with the brother did not replace the offering, but rather that it was to be done first, with the offering to follow.

We are no longer under the Torah (and those of us who are Gentiles were never under the Torah), so its application to us is that our Christian walk is futile if our interpersonal relationships are not what they should be.

this is actually exploring the entire theological framework of reformed theology using this particular verse only

That's a bad idea from a Reformed or any other perspective. To take one verse and attempt to understand a full theology based on it is simply not possible. The surest way to misinterpret Scripture is to take verses out of context.

Taken in context, this is not speaking about straightening out a specific sin prior to going to the altar, it's about the principle of going to the alter to "worship" with a heart that's far from God. It's a perfect complement to Hosea 6, which speaks of Israel turning to God with an unrepentant heart simply to achieve personal gain - the healing of their land. God rejects this, and in verses 6-10, we see the underlying principle:

6 For I desire mercy, not sacrifice, and acknowledgment of God rather than burnt offerings. 7 As at Adam,[b] they have broken the covenant; they were unfaithful to me there. 8 Gilead is a city of evildoers, stained with footprints of blood. 9 As marauders lie in ambush for a victim, so do bands of priests; they murder on the road to Shechem, carrying out their wicked schemes. 10 I have seen a horrible thing in Israel: There Ephraim is given to prostitution, Israel is defiled.

This has nothing to do with salvation, predestination, or any other Calvinistic doctrine. It has to do with the truth of the condition of our hearts. It speaks to the pointlesness of following religious ritual, and the superiority of a heart that is turned over to God. Such a heart will understand that love, kindness, and forgiveness are necessary, and will not worship in vain.

From Calvin's commentary:

It amounts to this, that the precept of the law, which forbids murder, (Exodus 20:13,) is obeyed, when we maintain agreement and brotherly kindness, with our neighbor. To impress this more strongly upon us, Christ declares, that even the duties of religion are displeasing to God, and are rejected by him, if we are at variance with each other. When he commands those who have injured any of their brethren, to be reconciled to him, before they offer their gift, his meaning is, that, so long as a difference with our neighbor is kept up by our fault, we have no access to God. But if the worship, which men render to God, is polluted and corrupted by their resentments, this enables us to conclude, in what estimation he holds mutual agreement among ourselves.

It's not a legalistic "do this, get that" type of thing. There is no "specific sin" to be taken care of first. It is, as Christ is trying to teach in this passage, a matter of hypocrisy vs. true Christian love, which is a result of faith in God and repentance.