I’ve been looking through the plugins and agree that Booking calendar looks good for this, especially as payment does not need to be taken.

The reviews look good (including changes from 1 Star to 5 stars because the devs are so responsive!) and there is a good base of people using it.

I haven’t used this myself but think this plugin would work. I imagine there would be some initial work to add current bookings in and providing your team log any rentals not taken online in there, removing the potential for double booking, it seems like a good choice.

Good luck!