Laptop BSODs - battery?

[9-Ball]

I have an odd issue with an HP Pavilion G6 (spec), where it keeps hitting a BSOD and then on rebooting runs an eternal Check Disk that either freezes 56% in or completes and boots into W7. The BSODs report:
STOP 0x0000004A: IRQL_GT_ZERO_AT_SYSTEM_SERVICE

I have done a clean install of W7 to hopefully rule out OS/driver problems. The laptop is not OC'd in any way. Obviously, this strongly suggests a failing SSD.

However, the weird thing is I can only get the system to crash/BSOD when the laptop battery is in the case. When I have the power cable plugged into the laptop and the wall, it won't BSOD, at least after 2 days of testing/idling.

So, my questions are:

1. Is it plausible a battery problem could cause such behavior on a laptop?
2. What's a good tool to thoroughly test the SSD to try to make it fail/BSOD without the battery, so I can rule the SSD out as possible cause.

Or, other thoughts on cause?

 

[BadAsAl]

Run Memtest x86 overnight to test RAM.
Run Seatools for DOS (start with short tests then do long test if no errors).
This will help you eliminate RAM or SSD as issues at least. If these pass then let us know so we can try some other things.

 

[9-Ball]

I have successfully run MEMTestx86 for 12 hours without any errors. Sorry. I should have added that in the first post.

I also ran CrystalDiskInfo just now and did actually hit one error. It reports:

C5 Current Pending Sector Count || Current=100 || Worst=100 || Threshold=0 || Raw Values=8

I'm guessing that points to sector problems on the disc after all? If ChkDsk has run on this about 8 times and not found/fixed it, where to next?