With the UV numbers on the frames and pictures, we can safely just discard what is in the photos. If there are numbers written on the pictures and frames, the pattern is clearly:
1. Pictures tell you the order
2. UV light/Sharpie tells you the number.
There is no reason to look for numbers in the photos when each photo can only have one number and we have 4 written numbers. If it was the number of people or the number in the photo or whatever, that would be consistent and we already tried that.
Ultimately, this puzzle was made to be solvable. If you need to make a leap in logic not contained in a clue, you're overcomplicating the puzzle because no one would assume someone would make such a leap when designing it.
We need to match the pictures back with their frames. The missing one might be the birthday and the clue on the envelope.