Is there a way to use the SFIV Tournament Edition Arcade stick on the PS3 if you have the Xbox360 version or vice versa besides buying the XCBM or whatever converter?
Unfortunately, no. They are not cross compatible. (Also the converter you speak of has been said to be pretty poor on top of that.)
Your best bet is to get a PS2/PS3 stick at some juncture.
That is terrible, I have to spend another $150 then for the PS3 version. They are all out of stock in store too.
You can mod it.
Or buy a Cthulhu and mod that 360 TE to be dual 360/PS3 compatible.