Lightning,
From what I have read about registering an imported US a/c in UK on CAA or FAA register, FAA is far easier. I do not know what you are intending to use the 337 for, but you could get an FAA IR and fly IMC. Of course, this would be much easier than getting the JAA equivalent.
Could someone please confirm that FAA C of A checks are less expensive than the CAA equivalent. That could mean big savings with a twin.
Finally, I have read that the FAA have a unique category of license type for the 337 due to it's single axis engine mounting. Any benefits due to this might be worth looking into.
Good luck.