How are you funding that lighting wallet? On chain tx must be made to open a lightning channel. Therefore there is no privacy unless one uses whirlpool to begin with everything else is not private but bunch of bologny.
Only Whirlpool allow you do break deterministic links to your bitcoin's previous history, and provide you with forward and backward looking privacy. When spending bitcoin after using Whirlpool (from your postmix wallet account), it makes it extremely difficult for anyone to gain information relating to your bitcoin prior to it entering Whirlpool. So if your onchain footprint is shit your lightning opening Tx will be linked to your id especially if one sends to open a lightning channel directly from an exchange wallet.
Did you even asked these questions or know about this?