If, he has a dustless system and air filtration system, he can cut inside. Is this site-finished or factory finished?

The contractor can use a system that will capture most of that dust. What is your biggest concern? Needing to clean the dust off everything afterwards? I can understand that.

Site-finish will create a mess, so why not allow them to cut inside?

The installers can set-up in the room, they are working in. They will need to move their equipment around. The saws can be connected to their vacuums. This will save them 1-2 hours of walking. The mess is contained in one room and not tracked throughout your home. Cleaning one room is a lot less work, than cleaning an entire house.

You did everything backwards if you finished all of your walls and woodwork before the floors. The floors come first. Now you are going to have to deal with moving everything out, lots of touch ups, and everything else.

No contractor is going to take the job if they can't cut on the same floor as the job. Unless you pay a big upcharge. You are not being reasonable.