2010 Mini S will not start

Good afternoon everyone,
So my wife bought a used 2010 Mini Sport roughly 5 months ago with 90k
miles on it. Around a month in it started dying on her randomly while driving.
Not frequently, perhaps once or twice a month while she was driving more
than a mile or two. She said it would click and then completely die. Not a
slow gurgle, just click, and no power. Dead. Any ways, we would normally
wait about 10-30 minutes and it would start up again and we could go, but no
sooner than that. I took it to a shop and it had no error codes or warning
lights, and it ran perfectly. They said they couldn't figure it out so they gave it
back to me. I would assume they just couldn't be bothered finding the real
problem. So far I have tested the battery and it is good, but was low on
charge. I also checked the ignition coils and they are all good. The only thing
I haven't checked, mostly because I am not sure how, is the fuel pump. Any
ideas or anyone have this problem that knows the solution?