@RichardStanley I am not doing the opposite, the join is above in the diagram. I don't understand if your comment is aimed at simplifying my argument or if it gives the proof required in the very last question in my post.

@RichardStanley excuse me, but with a union closed family aren't we generating $L$ with the join operation rather than the meet operation?

@SamHopkins thank you for your comments. I think that I managed to do the easy part, and can be satisfied by having shown that we can restrict the elements to only meet-irreducible ones. However, proving or disproving that this is a representation with a minimum number of elements seems more difficult.