Does it have to be that specific size? The reason I ask is because there's a good chance if you custom make it, it'll be much smaller which might help. But you do want to prototype it first, have all the bits and pieces in place that you will end up using on the final design. There are many parts for which you can get both a through hole as well as a smaller SMD version. So make a prototype with the through hole parts and when you know everything works for sure, then recreate it in a SMD form factor.
Going back to your original post, do you have a link or datasheet for the LED strips you plan on using?