TWO AND A HALF DAYS TO RENDER. AAAAAAAAAAAAAAAAAAAAAAGH. Probably could have done it overnight if I had my desktop machine, but it's more than halfway across the country. Wish I knew anything about how to tell if your render will be decent without having to wait several hours for the trial-and-error method. For this one, I simply resorted to brute force...
Splits-elliptic with Foci and MobiusN final transform.