Then it all goes terribly wrong and Paul kills himself by trying to change THAT much of himself. Also, J'aarkn gave his notes to M'gann, not Paul. Thirdly, OL kinda freaked when Jade said that she had changed her brain a bit...you really think he's up to changing his entire being? Honest, I don't even think the ring could, data wise, unless maybe Paul wanted that more then anything else in the world. Plus, the only thing that it would give him, that he can't have now, is Shapeshifting.
It's true that it's not going to happen because the SI doesn't see a Martian physical body as being more desirable than his current buffed-up human bod. I'm pretty sure the access to the data could be written off with a throwaway line like "I pull up the copy of Mr J'aarkn's data that I passively scanned in passing that one time", or even "Ring, collate all information observed about Martian shapeshifting and design me some exercises." As for difference in powers, the SI has already taken actions to make sure he's not reliant on the ring (or any other one thing) for some things, and there's certainly no reason to refuse to take on board redundant powers. Particularly ones which are species-inherent, meaning that even if the SI was stripped of his ring(s) and every other piece of technology, he'd still have those abilities. Bonus points for being able to do those things without draining ring charge.
The Renegade is probably more overt about this, with the Sword of First and Second, aero discs, a Father Box, the Venom Buster, and New God awakening meaning he's still fairly capable even without a power ring, and to a certain degree even without the discs and Box. The Paragon has his gun and flight belt, but more subtly the initial physical changes (and subsequent maintenance) made by the ring have still resulted in him being very fit and healthy, most likely able to keep pace with Olympic athletes. Not necessarily to a superhuman level, but certainly to peak human. His anti-scry tattoos, too, are very close in equivalence to implants more than clothing or equipment. And he actively worked to grow a soul he didn't have before; that's a fairly integrated component to add.
All in all, in any universe, he's not above altering his appearance fairly significantly, even as far as his apparent species. And a Martian physiology would allow him to look more human than Grayven currently does - or switch back and forth as he desired.
Alas, from a narrative perspective, although it wouldn't change too much about what he could achieve in a story, it does make it more difficult to temporarily depower him as a character in order to raise tension. Although it
is a DC universe, so alien tech or super-science capable of handling Martians is probably lying around everywhere anyway.
