The 2000mAh battery will fit using a thick back cover with either a CF adapter, or mSATA.
Using either of the above runs you into other considerations however. The CF cards are just too expensive in the high capacities, at least here in the U.S. they are. So out goes your "cheap" requirement.
The mSATA cards are more reasonably priced, but depending on how diligent you are in monitoring the battery level, you can end up with corruption issues, they aren't the most stable choice. I've had good luck with using the 2000mAh battery and the BP4e 480GB card. But I'm simplifying that scenario, bottom line is the mSATA option while attractive has it's quirks and drawbacks, it's a bit more of a high maintenance relationship, but quite rewarding if you put in the effort.
That might bring you back to SDXC in terms of your specifically stated aims. There is unfortunately, no easy or clear cut answer to your battery question. Everyone wants that easy answer, but it doesn't exist. Your run time with an 850mAh battery and SDXC will vary, depending on how often you illuminate the display, how often you navigate menus or browse album track lists, advance the track, and even right down to what kind of files you have. For instance, AIFF playback uses more battery than ALAC, which uses more battery than MP3, etc...
The best anyone can do to half figure what kind of run time they might get is to take a look at
Tarkan's page on run time shootout, so far the most detailed battery comparison I've seen. You'd still then have the added variable of which exact battery you choose, whether or not it is of good quality, or a fleaBay pack of lies in terms of the stated capacity. Cheapest is definitely not best in terms of battery selection.
If you purchase a Tarkan adapter, I'm sure he'd be willing to point you in the direction of a U.K. available battery of known quality, even if it is sourced from eBay, he seems to know who the reputable sellers are.