Go with the 111mm bottom bracket. Then, to get the magical perfect chainline, add a 1.7mm washer to the inside of the hub threads before spinning the cog on. But it's not that big of a deal.

If it were me, I'd spend a bit more than $6 for a bottom bracket. It might work fine for years and years or it might go to ***** in 3 months.

Be sure and grease your bottom bracket threads and the bb cartridge before installing. And every year, pull that sucker out and regrease.

Otherwise, THIS might happen.