There IS a standard for signing messages from any address type available, and it has been around for many years now and is recognized by major bitcoin implementations. If one software doesn't have the feature then it is that software's flaw and you should complain to their developers about why they haven't implemented such an old feature.
I don't think Bitcoin Core implemented either of the BIP? Or I can't find any of sign of the two being implemented.
Yes, it's possible to sign a message with the private key but you cannot refer to the bech32 address directly in doing so; exporting key-> sign message. You have to access the console to call signmessagewithprivkey and use your private key to do so.
As I said BIP137 existed long before it turned into a BIP and bitcoin core does support it (at least 9 years ago) it is just not called that and it doesn't support the SegWit part and it is bitcoin core's missing feature which core users should ask the devs to implement it. (but nobody seems to be interested in the feature enough to make the effort!).