I think a lot of model paints will fit your needs. Spend the money on nice ones though; paints save costs by reducing the tint. (Again, I’m assuming you’re not going with bluing. Bluing gives a beautiful color, but won’t do red and requires a different skillset.) You want one that can give a lot of color with a very thin layer. I’ve not heard of enamel being especially heavy among paints. But that’s also why you want to experiment. I would NOT do spray paint. If you have an airbrush kit, you can control the feed so it’s very thin. But spray paint will be too much. Instead you’ll want to use a brush, probably size 0 or 00.
Fortunately, getting the watch movement out of the case is super easy. As long as you have a VERY CLEAN WORKSPACE (including floor) you should be alright removing your movement. Number one frustration is one tiny speck of dust or cat hair gets inside and then you have to go pull it out to clean it again. I run the air filter, wear gloves, wear a hat, wear a mask, just to hopefully minimize dust. Other people don’t feel so compelled But re-opening and cleaning is easy.
I am hearing a watch repair kit in your future. They’re inexpensive and make life much easier. The DIY Watch Kit one really is good, although their shipping time to me in the US is forever. You’ll use the tools to remove the hands. It’s not very hard either, but worth practicing on your spare watch so you get it down. There’s a risk of scratching the dial, but it would be hard to really damage anything at this step.
Then you put the hands in your vice (careful not to bend them) and apply paint. Cover them to avoid dust and wait for them to dry. Obviously practice on your trash watch first.
Once dry, you’ll use the hand tool and follow the instructions on the DIY Watch Club youtube. Hour hand is easy, minute hand is easy, second hand is tough. I’ve seen people do it in thirty seconds, but you and I are not that skilled. Take a break before, move slow, expect it to take 30 minutes and that you’ll drop it on the floor a bunch. But forcing the second hand in if its not aligned correctly can break your movement. (I’ve put second hands on maybe twelve times, I misaligned it once with no damage. Just… be patient and don’t force anything until you’re sure.) Then put the movement back in and close it up.
This is really a fun hobby and it’s neat that you’re trying something no one has done before. You can definitely do this Just take your time.