Here’s my understanding: In speaking of our present, mortal bodies and our future, immortal bodies, Paul is using proskairos and aionios to contrast two measures of temporal duration in order to emphasize the superiority of our future immortal body over our present, mortal body. In Matthew 13:21, Christ employs proskairos to denote a relatively short measure of time. It is used in reference to those who hear the Gospel and endure in their faith for only “a short while” in contrast to those who, after hearing the word, keep it and go on to produce fruit. In this context, proskairos conveys a relatively short measure of time, the passing of which is observable by mortals. Aionios, on the other hand, while not denoting endless duration in an absolute sense, is often used in reference to relatively long durations of time. Here, it simply conveys the idea of a relatively long measure of time, the passing of which is unobservable by mortals.
So when Paul says that “the things that are seen” are proskairos (i.e., pertaining to a relatively short measure of time, the passing of which is observable) while “the things that are unseen” are aionios (i.e, pertaining to a relatively long measure of time, the passing of which is unobservable) he places our present, mortal bodies in the former category of things, and our future, immortal bodies in the latter category of things. Being imperishable, our resurrection bodies are not the kind of things that pertain to a relatively short, observable measure of time (like our mortal bodies). This need not mean that Paul understood aionios to denote the entire duration of time during which our immortal bodies will last after we have been clothed with them; it was simply the most appropriate time-word with which to contrast proskairos in order to emphasize the imperishable nature of the spiritual body over the perishable natural body. Though aionios does not denote endless duration here (or anywhere in the NT, I would argue), it does serve to forcibly emphasize the superiority of the resurrection body over our present bodies.
That this is Paul’s meaning seems evident from the next verse (5:1), where Paul speaks as if our resurrection bodies are presently waiting for us “in the heavens.” In speaking as if they are already in existence (though in actuality they do not yet exist), he calls them aionios (again, which pertains to a relatively long measure of time that is unobservable) to emphasize their imperishable nature over the perishable nature of our mortal body.