I'd probably get the later Series III. More modern updates to the car and probably in better condition. I'm not aware of any major differences in them beyond cosmetics and more modern features like ABS and A/C etc.
I've been thinking of the same project for the last year. Taking a nice XJ6 and dropping in a more modern, fun and efficient power plant. The most obvious option is some kind of LTx or LSx motor and a T56. Although a BMW powered option might work also. So might something out of a 350Z or Infinity.
The front suspension on the car is pretty straight forward dual control arms. The rear is complex with lots of rubber bushings to provide the smooth ride the Jags are known for. They make poly bushing kits for those but the brakes are still in-board. If I read the suspension diagram correctly there are TWO shocks on EACH side of the rear suspension! I wonder if a rear subframe swap would work. Maybe something out of a BMW, RX8, Mustang etc
Konis and Bilsteins are readily available. Performance springs and sway bars seem rather expensive - probably due to lack of much demand.
The main issues that I've read about from my limited research appear to be electrical thanks to the Lucas electronics so switches and alternators and such can be problematic. Interior repairs can be expensive as well if you are looking to keep the original style interior.