Ahh, if it's going to be from Wodehouse, can't be Jeeves, but more like "Bertie Wooster", the high-spirited but feckless one who always needed rescuing.
I'm thinking the creature is female, though, so I'm going with "Inconstance".
Having previously owned a 52 MGTD replica, a 68 MGB, a 76 MGB and a 74 MGB, I recommend the name Morris, after Morris Garage, sacred to MG history, which, with Lucas electricals (Sir Lucas, Prince of Darkness) and the British penchant for autonomous leaking of fluids and mechanicals going out of adjustment, is exactly where Tommy's MG TD will spend its time.
I agree about it needing a British name, and those headlamps look like spectacles to me. I'd suggest Reginald--not Reggie, which is way too sporty. Sir Morris was a nice suggestion, too. Or Maurice (which the Brits pronounce as Morris)?