Do you already try to use the Atomic wallet, Jaxx Liberty wallet, Coinomi wallet, or another multicurrency wallet to swap your bitcoin to ltc? That can help you to stay away from KYC, which you don't want to fill. You can use an online service such as Changelly, switchere.com, coinswitch.co, evonax.com, etc. But I never tried with the online service because I prefer to use the wallet for my mobile phone or my desktop.
I've tried those services except those that I've crossed and I'm satisfied, I never had problems with them and OP said the exchange on Exodus, the swap. I have used it too and never had problems as well.
You now have a lot of choices OP.