Is modern Evangelical Protestant soteriology consistent with the early Christian idea of "theosis"?

Theosis is really more of a non-issue.

Comparing two faiths with (more or less) the same scriptures means that this is going to have some extra-scriptural material. These are some well-researched, thoughtful articles on the subject.

Becoming like God: Am Evangelical Doctrine of Theosis

The idea of divinization, of redeemed human nature somehow participating in the very life of God, is found to a surprising extent throughout Christian history, although it is practically unknown to the majority of Christians (and even many theologians) in the west.

This paper is supportive of theosis being compatible with evangelical Christianity.

While there are weaknesses in theosis theology, the strengths are considerable. The doctrine of divinization merits the ongoing attention of Scripture scholars, theologians and pastors who desire to provide significant resources to Christians in their quest to become like God.

For a good response, read A Case for Progressive Sanctification: Responding to the Doctrine of Theosis in Evangelicalism

The conclusion is that evangelical Christianty could be construed to be compatible with or believe in theosis, but its really comes down to what you mean when "man becomes like God".

If theosis affirms more than progressive sanctification, yet does not make God human or unite God to sin, and does not deify humans (theosis) ontologically, then we should use the term "progressive sanctification" instead of "theosis" since this theosis is in name only.

Evangelical Christianity does not really address theosis. The word just isn't in the vocabulary.

It's can technically be compatible with some interpretation of it, but it would usually be a misleading description.

My short answer is yes, they are compatible but you're correct to observe a shift in emphasis.

The reason for the divergence in eastern and western thought is that in the Bible there are a variety of different analogies and explanations for the nature of salvation. The west at some point latched more onto the juridicial/courtroom analogies, while the east latched more onto what Adolf von Harnack described as "mystical" explanations. Donald Fairbairn, an evangelical and a professor of patristics, analyzes and (to a degree) critiques this narrative. Says Fairbairn:

The rise of the penitential system and the notion of the Mass as a re-sacrifice of Christ at the beginning of the Middle Ages, the Anselmian revolution in atonement doctrine in the 11th century, and the crystallizing of the sacramental system in the 12th–13th centuries all served to solidify this juridical soteriology [in the west], in which salvation was seen as a state before God. Furthermore, it seems to me that the Reformers did little to reverse the juridical trajectory of the Western Church. Granted, the Reformers gave a radically different answer to the question of how one attains a right standing before God, but most of them did not fundamentally alter the general idea that salvation is primarily juridical in character.

Elsewhere in his article, he emphasizes that, for many of the fathers, the idea of theosis was not so much sharing God's qualities but rather sharing in the fellowship between the persons of the trinity. This idea, while not wrapped in the language of theosis, is huge in the recent evangelical book Delighting in the Trinity.

Athanasius is often regarded as giving the idea of theosis the best and clearest expression in the early church. Analyzing his writings on the topic, Philip Edgecombe Hughes defines Athanasius' concept of theosis as:

the reintegration of the divine image of man’s creation through the sanctifying work of the Holy Spirit conforming the redeemed into the likeness of Christ, and also of the believer’s transition from mortality to immortality so that he is enabled to participate in the eternal bliss and glory of the kingdom of God. (P. E. Hughes, The True Image (Grand Rapids: Eerdmans, 1989) pg 281. Quoted in Robert Rakestraw, Becoming Like God: An Evangelical Doctrine of Theosis)

So defined, the doctrine is compatible with various evangelical soteriological frameworks. Ultimately, the doctrine of theosis gets its support from the Bible. There are a number of passages speaking directly or indirectly of it, but 2 Peter 1:4 is the best place to start:

And because of his glory and excellence, he has given us great and precious promises. These are the promises that enable you to share his divine nature and escape the world's corruption caused by human desires.

Throughout history, though the western emphasis has been on juridicial explanations for salvation, recent scholarship says deification language isn't entirely absent in the west historically:

Many assume that the patristic notion of deification is absent from the mainstreams of post-patristic Western theology. Recent scholarship, however, identifies deification in Augustine, Aquinas, Luther, early Anglicanism, early Methodism and Jonathan Edwards -- all fountainheads of Western theology. This article contends that deification is also present in Calvin's theology. It is not a prominent theme in its own right and some of the bolder patristic terminology is not employed. Nonetheless, the concept and imagery of deification regularly appear on stage while other doctrines are explicated. For Calvin, deification is the eschatological goal and blessing greater than which nothing can be imagined. (Carl Mosser, The Greatest Possible Blessing: Calvin and Deification, Abstract)

I write from a reformed perspective so Calvin is the most significant of the bunch for me. In reformed thought the terminology of union with Christ is often employed to express this idea. The seventeenth century work The Life of God in the Soul of Man is a classic on the topic.

Really, what "theosis" expresses is the idea of sanctification (becoming more holy and more like Christ) and glorification (being finally so like Christ as to radiate with glory) wrapped into one. The reformers emphasized the distinction between these things and justification in order to emphasize that our union with Christ, while synergistic, is not possible without God unilaterally (monergistically) declaring us righteous first. Orthodox and Catholic thought don't make this distinction. Therefore, in Protestant thought, salvation and justification are often equated and therefore spoken of juridicially, even while recognizing that the goal of our justification is to reflect God's glory more and more.

While the language, and even a lot of the underlying theology, may be different between east and west and between Protestants and non-Protestants, the idea of becoming like God in our salvation is not foreign to evangelicalism. John Piper, an evangelical pastor, briefly touches on this in his spoken biography of Athanasius (though it doesn't seem to be in the abridged transcript).

The goal of entering God's kingdom is to enter into fellowship with him.