How do believers in the tripartite nature of man (body, soul and spirit) reconcile Genesis 2:7, 1 Corinthians 15:44-45 and Matthew 10:28?

The fundamental issue here arises from the fact that contra to your assertion, the biblical meaning of "soul" is not straight-forward. Sometimes the Bible appears to use the word to describe a whole person (cf. Gen 2:7, 1 Pet 3:20) - which is the emblematically dichotomous view - and sometimes it appears to describe part of them (cf. Matt 10:28, Heb 4:12) - recognising the latter is the essence of trichotomism.

The resolution to the problem you pose is to recognize that when words are used in different contexts, they can mean different (if closely related) things.

From a trichotomistic perspective, the weak leg of your argument, is your first premise:

P1: a living soul is a combination of physical body and breath of life (from Gen 2:7 and 1 Cor 15:44-45)

The Genesis account describes the formation of the first living soul, but to say that what is there entirely defines a "living soul" (let alone "soul") over-eggs the pudding.

Consider the following analogy:

If we describe human conception in the following terms - "A new human being is formed following sexual intercourse when a (male) sperm fuses with the (female) ovum."

If we then proceed to claim that this necessitates that we define human beings as sperm-ovum fusions then we have gone too far. What's happening here? Well, we haven't taken in to account sufficient data like the fact that as soon as mitosis (cell division) begins, we no longer have a simple sperm-ovum fusion, but something more. This is more evident in the case of identical twins - here, the definition fails us completely as it doesn't suggest the presence of a new human being and if we stick rigidly to it, we will be left with absurdities.

Trichotomism is not readily formalisable to the level of definitional clarity that you are seeking, as it is not a formal philosophical theory built up from defined terms and propositions, rather it is a synthetic biblical doctrine based on the underlying premise of the harmony of scripture.

To call the issue you have raise a paradox (let alone a contradiction) is forcing the use of the term, as you are relying on a definition that is necessarily incomplete (otherwise there would not even be a hint of contradiction).

There are mysteries here that will most likely remain obscure this side of glory, but to this trichotomist, the question, while not definitively answerable, upon the examination of all relevant biblical data is suggestive of the doctrine of "soul sleep" (cf. 1 Cor 11:30, 15:51): Upon physical death, the soul neither ceases to exist, not is it correct to call what remains "a living soul"; our spirit is certainly not destroyed, but for it to manifest it's individuality to an extent akin to (but beyond) a "living soul", it will require a resurrection body.

Perhaps another analogy is in order, think of the temple/tabernacle: The "spirit" of the temple/tabernacle is defined in the Torah - the essence of the system; The "body" consists of the constituent components of the functioning temple/tabernacle - the actual structures and the actual artifacts but also the priests and sacrificial animals etc.; The "soul" would be an instantiation of the system (eg. wandering tabernacle, tabernacle pitched at Shiloh, in Jerusalem, 1st temple, 2nd temple, 3rd temple). In one sense, if you destroy the "body", you also destroy (in some sense - at least according to man's view) the "soul", but (if you believe in the Redeemer of Israel) not finally - it can be "resurrected" and from a God's-eye view, future instantiations are observable realities unless there is a true "casting in to hell".

In answer to the (commented) question "Are all trichotomists 'soul sleepers', or is it just a personal position of yours?" - I don't think that is necessarily the case, but in general, people who analyze things based on a given set of premises and the same data will tend to come to similar if not exactly the same conclusions - so, although I don't have any hard data either way, I would not be surprised if that is the case.