Congratulations on your upcoming birthday. A 1960's luxury watch sounds like an excellent way to honour this occasion. I have a birthday coming up in June, as well but my wife says that I already have too many watches. I respectfully disagree on this matter but she says that I don't know my ass from a whole in the ground.
I have a very small collection collection of cars but I don't I have any E-Type Jags. If you are at all open to doing a trade, we could both celebrate our birthdays without any impact to our bank accounts. I don't have many watches that are gold capped but I can probably find a solid gold one that would meet your approval. If you have an attachment to this vehicle, I completely understand and will begin my search for a birth year E-Type.
You will have figured out by now that watch collectors are, in general a fairly obsessive bunch and there will be zero encouragement, or positive feedback towards any watch that you find with a dial that has been repainted. As a car enthusiast, you will no doubt appreciate the difference in value between a vintage classic that has the original paint, compared to one that has been refinished. We want the original dial and hands. Many of us appreciate the signs of age, or patina. For a smaller group of us, if some of that paint on the dial happens to be radioactive and glows in the dark, all the better.
The E-Type Jaguar is a legendary design that is right up there with the all time greats like the Mercedes 300 SL, or the Cord. As an aging car enthusiast myself, I will offer my thoughts. Your birthday warrants something that is more in line with your choice of vehicle: elegant, authentic and reflective of the design language from that era. Forget gold capped, as these watches were for people who liked the appearance of gold but could not afford it. You can afford solid gold in the price range that you are showing us. A simple, 3 hand, solid gold, manual or auto Omega or Longines can be found with dial, hands and case in excellent condition. Forget black, unless you can increase your budget, as it wasn't all that popular and did not age well. A gold Constellation will be considerably more expensive but if you are open to spending a bit more, IWC watches from the 1960's are also beautiful and would align with the status of your car.
Good luck with your search and if the trade is an option, here is an example from my collection.