Am I am still allowed to travel?
Won't be a popular answer, but WTF.....
It sounds like it's probably a rare type of car - but you haven't mentioned what it is. If I was fortunate enough to own a Maclaren or similar, then I certainly wouldn't be letting any local garage get their hands on it. A specialist car needs specialist mechanics, and if that is really what you are describing here then I would be taking it the 150 miles. But I wouldn't have asked a random bunch of cyclists on the internet first, surely you could have predicted the verdict? Especially on the board you chose to post on. Try the motoring board, they might be more sympathetic.
To my mind it is no more risky than the huge amounts of van and lorry drivers that are still driving all over the country very day as part of their job. Don't tell me they are all being tested frequently, or adhering to the rules. The roads around here are still very busy, as they have been throughout lockdown mk3. I would suggest you do all the obvious stuff like fill with petrol locally before you leave, carry food and drink with you so you are minimising stops and contact with others. Wear a mask, use hand sanitiser. You are then doing more to prevent spread than most of the numpties I see coming to my home town for no obvious reason. Go straight there, get your MOT/service done, and drive straight home. Use common sense.