@Kenzie84
You can't rely on the calendar synch always working - as you have discovered - even when it does work there is often a lag - potentially allowing for double bookings. Some hosts always carry their phone with them, so that when they get a booking via one platfrom they can quickly block it off on another _ you can still have your calendar synched - it doesn't hurt to double block as it were.
sounds like you have IB - in which case you can get 3 free cancellations a year.
However on the basis on your listing saying your max is 4 but the guest want to bring 7 - I'd be ringing CS and askign them to cancel on the basis that it is violating your maximum occupancy - (you are on shaky ground however if you have told the guest in writing via the airbnb platfrom that you will accomodate them)
Your third option is to tell a bit of a fib. Contact the guest and say your insurer will no longer allow you to exceed maximum occupancy. and even if they turn up, you would have to turn them away. Tell them they will need to cancel and if they dont' cancel soon, they will lose the ability to get a full refund. I"d try the 3 rd option first. Dont' mention the booking via the other platform to anyone.
Other hosts may have better ideas