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Is there such a thing like a Lightning Proxy? Like a custodial service that can receive Lightning payments and forwards to my real address?
Yep. I think this is what you are looking for: https://github.com/lnproxy/lnproxy-address
But in this example you would need to self-host it, so not really custodial. AFAIK, using Zeus with their lightning address offering would achieve the same result, as I'm pretty sure they wrap your invoices (giving the privacy you are looking for).