I don't want to come off as self-promoting, but the DnD DM problem is exactly what we're working on solving. If you build the infrastructure around the ruleset and leverage the AI for the story telling then it becomes much better than the vanilla ChatGPT experience, i.e it can reference your character sheet, ask you to roll for skill checks, do combat, etc.
It's far from perfect but we're continually working on improving it and you can check it out here if you're interested: https://www.fables.gg/
My friend and I had the same idea and ended up making a discord bot that manages all this for you. Can check it out in our discord if you're curious: https://discord.gg/nK7J4Jcg