The only two 1-dimensional representations of the symmetric group (i.e., what you can call fixed points if you squint a bit) are the ones that correspond to the partitions (n) and (1,1,...,1), which are the least interesting ones. Everything else is significantly more complicated. It's a block-diagonalization (Young seminormal form) with lots of nontrivial blocks. So no, you won't understand it in terms of fixed points (and certainly not of Lawvere's theorem, which I've never seen used in the entire subject).
I knew Craige Schensted left his mark on the subject; I had no idea his wife (Irene Verona Schensted) had written a book about it! (well, I think it's about it; it's not on libgen...)
I have a linkdump on this
https://github.com/adamnemecek/adjoint
I also have a discord https://discord.gg/mr9TAhpyBW