The internal dynamics of Christianity are against it, but it would certainly be interesting. Without its privileges and wealth, the church will be smaller and less significant both socially and intellectually. The Roman world will need to come to terms with cohabiting, mutually exclusive absolute truth claims, whjich is quite a challenge (though something both China and India managed, so not impossible).
There will be churches, but not basilicas, bishops, but not archbishops and ecumenical patriarchs, and probably neither monasteries nor prebends. The most likely structure is one of religious collegia or corpora, with the funds under the management of bishops. They'd be subject to seizure, same as other temple funds, in times of need. Bishops would also need to be approved by the authorities, of course.
Theologically, it would make interesting things possible. In a world where not everyone is Christian - or supposed to be - the Church needs to decide whether to give up integration into wider society or its exclusivity claim on the lives of its members. I'm guessing the exclusivity will go, and in a few centuries it will be possible to be Christian, Stoic and Aristarchian simultaneously.
That is, of course, assuming the Church stands by the terms of the Gallienian Peace and no other exclusive-claim faith becomes *church.