Have you tried the Builder Profile under Settings & Beta yet, and linked your website to that profile?
I haven’t had time to try that out yet to see if that allows more robust sharing possibilities, however as it stands now, only people who can already access GPT builder can view other custom GPTs.
Otherwise, you would be looking at using the Assistants API.
Maybe it would be possible to get an external spreadsheet to ask it questions and retrieve answers using something like Zapier, then having that spreadsheet send the results to the public place where the question is being asked?