I’d rather do it myself.
I’d rather have done it myself.
(The Cambridge Grammar of the English Language, p.108)
Is there any regret in the second sentence for not having done it, or is it neutral?
Regret is implied.
The would have … done construction implies a counterfactual—for some reason you could not do it yourself—and the rather states that you wanted to do it yourself. Consequently you must have been disappointed in some respect.