The Romans ability to stretch further than the Hedjaz, you'd need two major things IMO.
1. To have the Arabian hinterland as your friend doing the actual fighting for you. I don't entirely know how to sell that alliance to be frank, rule over the Persian parts of Arabia? Not impossible, but still a hard fight, even with the Romans fighting in Mesopotamia.
2. Naval Projection - either allies in Somalia that have experience in sailing in this way, or basically a whole new endeavour as to my knowledge, their fleets never stretched far outside of the Red Sea at best. Now I think this is a great region for Roman Alternate History, with interesting sites for "Constantinoples of Africa", and the like, but fundamentally, it is an expensive endeavour that would primarily exist for attacking Persia in its flank. It is roughly as hard to control as Britain because of the distance, which complicates matters.
If we take that the idea that Romans bled gold to India as rebutted (as Pliny apparently said the Ganges was one of the best SOURCES of Roman Gold), then Trajan could as an aside, set up a fleet early in his reign to secure that, enabling the fleet to be used later on to push further around Arabia, allowing him to go further than clientelising Parthia, but also the Gulf.
Neither are easy, and you'd have to hope that the Romans can siphon enough taxes and tolls from trade in the Persian Gulf and Indian Ocean to pay for this fleet.
However, I reckon after Trajan, you'd basically lose the entire region. Either to rebels, or to the defacto independence of the Red Sea Fleet and Arabians. If you have the Empire divided, I can see Arabia being a seperate part, likely led from modern Yemen.
But overall, yeah - I could see it being a fast-track to Christianisation, butterflying away Islam entirely. Heck, it is early enough to butterfly away Christianity being embraced