Heat is the red part of the ring around the bounty marker, bounty size is the white part of the ring. They should talk to you when you don't have heat.

It's entirely possible to function while dodging guards, so it's not a requirement to raise an alt. Though if I were you I would very carefully choose where I am going to buy my mount, and use a place where I can nearly guarantee a clear path between the outlaw refuge moneylender and the stable master.