[impendia]

I am a professor at a Ph.D.-granting mathematics department in the US, so I can speak for the "other side".

Please, please do NOT suck it up for three years just because you are afraid of pissing off your advisor. That isn't worth three years of your life. Hell, it isn't worth three days of it.

If you were seeking a postdoc then, yes, a rec letter from your advisor would be absolutely required. But I presume you intend to seek non-academic employment? In that case a letter from another faculty member who knows you (and who knows your work) would fit the bill.

If you have an alternative career path in mind, why not apply for jobs now (i.e., without quitting)? If you find one to your liking, then you shouldn't care whether or not your advisor gets angry. If you don't find a job, or don't yet know in which field you would like to work, you can bide your time while you do some research on the job market.