To use BTCPay from Umbrel, you will need to use this guide to make it “visible”.
Or just use it as it is, but ONLY through Tor network. That means people will be able to browse your BTCpay shop ONLY if they use Tor browser (I hope you understand what that means).
Can you define what is “total anonymity” for you?
Please consult the guides section of this forum for further questions.