So, this rotawoman has been driving your children in her car since the September, last year (beginning of the current school year) but only now has told you about the 'disrespectful name-calling'?
Assuming your children have been calling her by her first name for many months, why has she waited this long before telling you? Why did she not straighten out this so called 'biggie' with you and your children in the first few weeks before habits got ingrained? It can't be that much of a 'biggie' can it! You could always try asking her (nicely) why she didn't tell you her preferences when the arrangement began.
Agree with pinotgrigo that her veiled comments about your children are ruder than your children calling her by her first name. If she has something to say to you about your children, she is not going about it the right way. And you also say things have been getting edgy over timings and other things, too.
I could be wrong, but it sounds to me as if she might be finding excuses for this arrangment to end soon. I think you should look around for someone else asap, just to be on the safe side.
But as you say finding a replacement is difficult mid year, I think you are right in deciding to respect her wishes for now, and gritting your teeth about her.