Battery was swelling so I ordered a new one and started to replace it. Screws towards the center of the laptop came out fine but the ones towards the edge are way too tight, possibly due to the swelling or due to manufacturer error; I've now basically stripped the screws at this point, they weren't coming out anyway. Is it attached to something below that I can use to remove it? Or should I just send it to Dell and take the monetary hit? I don't know how much 'you tried to fix it yourself' tax they'll hit me with, seeing as its out of warranty and I definitely know they won't put the battery that I ordered in it to save me the money. I don't have any fancy tools, by the way, just pliers/screwdrivers/basic stuff.
You need good quality screwdrivers that are just the right size for these screws. I have Phillps screwdrivers that range from #0 to #0000 for thsee screws. The manufacturers also usually use thread lock so they don't come loose by themselves. That makes them harder to get out but I have never failed to get them loose.
You might try a PC repair show first. It should be possible to drill out the screws (remove the heads) and leave enough screw body to extract them. They make screw extractors, but I've never seen any such thing for screws that small. You'll have to find someone with the right tools and experience to do this job. I really doubt Dell would do that. They would likely sell you a new case.