I'll show you what is popular with sign shops. I'm not sure what the coat is but you can go to the UL SAM and get an idea of the choices that are out there.
http://ul.com/global/documents/offerings/industries/lighting/signs/manual/SAM_October_2011.pdf
If you go to page 52, the GE Leds are very popular. You can select what colors, spread, brightness, etc. And order them through a sign supply house.
You have to get the transformer depending on your number of feet that you will need to outline the letters, and you can splice them from letter to letter using a small gage wire and wire nuts.
I would not worry with the polycarbonate because it would involve cutting that stuff which is not fun. Just peel the adhesive tape off and stick them to the inside of the letters. You'll need to make stand-offs to support the letters, but I would ditch the wood and keep it all metal to avoid the fire concern.
I think they would look cool with a blue light or whatever would fit with your space. Be sure to put your transformer in an electrical enclosure to be safe.
Good luck with your project! You'll be proud of it and you'll see why signs are son expensive!