How would the scarlet items effect work in the following scenario?

I was under the impression that joker items such as scarlet items counted towards the highest level set,

This wasn't always the case (but it did seem that way as higher set IDs usually correlated with higher levels, combined with the scarcity of joker items), as I distinctly remember testing with a Chaos Vellum hat long ago, the tempest set was selected over the empress set as it had the higher set ID. Regardless, it is apparently the case now, and set ID is used as a tiebreaker.

regardless of whether or not there's a ring or shoulder in the set

As everyone else has said, this is definitely not true, they will not count for anything outside of their own category.