What's the widest two-by size for a door frame I can use without warping becoming an issue?
You are correct for the most part if buying what is in stock at big box.   However if you have a local lumberyard that can get you something comparable to kiln dried SPF, I have never had an issue - or maybe 1 issue in 20 years.   

I do recognize your statement though because if I do buy 2x10/12s from big box half of them will go back with a few days - I way over buy because of this.   If you are worried about them warping after install then you should cross block the gap every foot or so and brace it in.   

I have had to do many many doors with 10by lumber.   I just make sure I buy good wood from lumberyard and get it installed within a day or two of buying.   I am not sure really what alternatives you have because I am assuming the door is load bearing.  I mean you could go with an LVL that is rated for vertical install and have them rip it to 10" - this seems to be way way overdoing it but just giving you a "perfect" option.