I'm not sure what DNA analysis would get you; maybe some sort of likelihood the body was that of a Jew. Carbon dating might get you the age to within half a century or less. That could prove it was St. Paul or at least a contemporary. But (speaking as an ex-Catholic) I can't see the Church allowing destructive testing on the possible relics of a saint.
No, the Church would not allow that, just as it cannot permit willful desecration for the sake of curiosity.
But you can see here how the mind of a non-Catholic thinks. There is this material thing here, and it belongs in a grave, and it's gross, so only someone really gross would want to touch it or look at it or move it or talk about it, unless, of course, there is some materialistic (scientific) motive. Therefore Catholics are morbid and gross. Period. Let's talk about something else. All of this precludes the reality of the spirit world. The Catholic interest in the relics of the saints has nothing to do with any such material or scientific basis, but is firmly rooted in the tradition handed down to us from apostolic times, beginning with the martyrdom of St. Stephen, whose mention is appropriate, for he was slain by stoning under the supervision of the same man with whom we are concerned in this thread. In the traditional calendar there is a feast day for the finding of the body of St. Stephen. The first Masses were said over the remains of such martyrs in the Catacombs, and even today the altar stones required for Mass contain a particular arrangement of the various relics of saints, all based on that tradition. As an "ex-Catholic" you ought to know all this. Which reminds me, when was the last time you attended a Traditional Latin Mass?
If anyone has any doubts about how objectively important the physical remains of the saints are, I would ask them to then please explain how satanists or witches are almost always able to tell from a distance the location of relics, and sometimes even the identity of the saint. One thing is certain: in exorcisms, the devil can be commanded to say whose relics are present, and therefore the devil always knows. But being a liar from the beginning, if you ask him (outside of the exorcism's binding venue) he would likely deny it.