d'oh! I had turned off the etymology tab on my OED! Now it makes more sense, and yes, under get it has the second listing:
2. a. What is begotten; an offspring, child. Also collect. progeny. Now only of animals.
b. orig. Sc. and north. In contemptuous use = brat. Also spec. a bastard; hence as a general term of abuse: a fool, idiot. (Cf. git.) Now dial. and slang. c. OED

So the sense transition from progeny to bastard to a general term of abuse seems quite plain (compared to some other transfers over time!)