When you say cheap batteries......how cheap (netto/poundland/lidl)?
Duracell Procell's & Energiser Ultra are a popular 9V choice, but basically any branded half decent battery should do it.
As the only job that the grip is doing, is to power a solenoid and drop a sear, i'd be very surprised if a Procell didn't last a weekend?
Ps. Always take the battery out overnight as the battery will drain down, even when not in use.
Alot of the cheap batteries are exactly that, and hence not worth it.
I've used rechargeable 9V's in my E2 cocker while practising but i wouldn't use them in an important game as you find that they die without warning.
In my opinion Lithium's would be wasted in a paintball gun, expense Vs duty cycle, is not worth it.
Stick with a half decent battery and you should be fine.
Do you have the standard WAS board in there? If so, that's half of your problem. They drink power - the APE board is better, as it starts on a trigger pull rather than having to be switched on with an allen key, and goes into idle mode straight after firing as well......rather than the WAS that just stays on for 10mins at a time. With an ape board, the batteries will last about 3-4 times longer.
Or you can get a new style Egrip, which are like the X7 ones, with the battery compartment at the back so you don't have to remove the trigger everytime you want to change a battery, and the board on it is very similar to the APE in terms of function. I have a brand new one for sale btw