Do people sign up using Calendly or via Airtable/form? You can set pretty specific availability rules right in Calendly. I know that “Maximum allowed events per day” is in the advanced availability options, but I don’t think they have a weekly equivalent.
I have a personal automation of Calendly to Airtable, just to track client meetings. I’ve also built a simple booking equivalent in Airtable, which worked fine but would be a fair amount of work to fully match Calendly’s features.
To do this from Airtable, you’ll need a way to count the number of meetings in each week, most likely using another table called Weeks. It gets fairly complex after that though, as you’ll need an automation to link it to the appropriate weeks record, a rollup to count the current bookings per week, and then some way to remove availability if that week’s bookings is equal to or greater than 3.
Personally, I’d recommend using Calendly’s features if possible, since you’re already using it, and then automating data into Airtable if necessary. I’d be happy to offer more advice, but would probably need to see more of what exactly the workflow is with Calendly.