I have been thinking about a similar approach to this as you describe above. An additional step you can take -- If you really want to reduce trust in any one LSP -- is to do a few hops. From Phoenix --> Breez --> Blixt --> <your preferred/cheapest loop-out option>.
A particularly compelling instance is if your KYC'd exchange offers LN withdrawals. The majority of centralized exchanges operate account-based systems for users, and so UTXOs are only "created" (associated with users) in an on-chain withdrawal. Thus, leveraging the LN withdrawal + multiple hop method obfuscates a user's future activity from all but the final LSP.
One or many LSPs need to have some level of trust in this strategy, but at least the user can determine that.