I'm honestly starting to get a little concerned with how thoroughly this (currently theoretical) weapon is being examined and designed, but I'll play along a little longer and consider these new iterations as well.

Lighting the stake on fire... eh. Might set the beast on fire, but given the way combustibles generally work it wouldn't exactly "burn them from the inside"; rather, the part of the stake ending up inside the beast - and most likely any part poking out of a potential exit-wound as well - would likely be extinguished due to being suffocated by the beast's flesh (since fire requires oxygen). It still might ignite the exterior of the beast or at least burn it some, which I suppose might panic it, but probably wouldn't be hugely effective at hurting it. You would also effectively be burning your stake every time you fired it (unless the stake was made by something fire-resistant but covered in something combustible, I suppose, though this would raise questions of just what this material would be), which would make it even more expensive to use since every stake would only last one shot.

Adding an explosive to the missile presents a slightly bigger problem than lighting it on fire, partly because you are still destroying the stake with every shot, but it also raises the question of how you'd actually manage to make it explode, and if you could do that, how to time the explosion right. Most things in Bloodborne detonate on impact, with the one exception being the delayed molotov, which works on a timer. I suppose one could set a timer on the stake-explosive, but then you would have to either set a timer on a case-to-case basis based on how long you think it needs to wait or, more likely, stick with a pre-determined timer that you just have to hope fits.
How to actually cause the explosion is a much bigger problem. The only things in Bloodborne that [I]really[/I] explode are cannon balls and (for reasons unknown) the stake driver and boom hammer, and considering that cannons fire using quicksilver I'd theorize that their explosive power actually comes from the innate arcane power released by the union of blood and mercury, and who really knows what goes on with stake driver and boom hammer that lets them explode over and over again. Then there are a bunch of arcane explosions from "spells" and, finally, molotovs.
Here we run into a very important problem: Bloodborne does not depict fire as fire. All fire in Bloodborne, as opposed to the real world, cause an instant burst of damage with no lingering effect, modelled as an explosion. I don't think it should be like that in the RP. Molotovs work like that in game for gameplay reasons, but in reality it is fire. Looking at molotovs in isolation I would be willing to concede that they are merely mislabelled and should explode, but their interaction with oil urns makes it clear that it is [I]fire[/I], not an explosion.
So basically, we would need to introduce something new to the world to act as the explosive for the stake... because nothing currently in Bloodborne (aside from "because Powder Kegs are crazy") explodes.
EDIT: You could just use gunpowder, of course. That's... yeah. Somehow I didn't think of that before reading "Powder Kegs" at the end there, simply because gunpowder isn't really used in the game.

By the way, if anyone needs help with their characters, feel free to ask.