Yea you have to do it properly
Swap back to Bitcoin broken into different amounts and at different times. Don't consolidate after.

-Cashu: Strong choice if you don't care about custodians for larger amounts sure. I would be worried about anon set of the mint though.
-Lightning: Maybe could be decent choice if you dont mind dealing with complexities and painful UX (you don't get the full privacy possible on lightning without running your own node. Custodians and LSPs sacrifice at least some privacy).
-Coinjoining: I wouldnt consider as a strong substitute since it is not only weaker obfuscation (nothing is hidden via encryption like the other options) it is also recorded on chain.

"If so, why not just use those instead of abstracting farther away from your SOV asset?"
Doesn't SoV imply long term? Swapping would be short term stuff. It shouldn't significantly affect the price at all. You could even make a tiny profit at these time scales from swapping.