The Eberspachers/Airstreams these are copies of can be run in fresh air mode or recirculating mode, so I can't see why these can't
Patrick's and mine run in recirculate mode - the air taken from the vehicle is cool enough for the shutdown cycle
Personally, I would consider extending your exhaust - the fitting instructions for the Eberspacher says that both the exhaust and inlet to the burner box MUST be extended so the burner inlet doesn't draw exhaust in and upset the burning cycle, and also so the fresh air inlet doesn't draw exhaust in when in fresh air mode.
The length of the exhaust and burner inlet should also be the same length so they are balanced.
Or, you could just leave it and see how it goes
Cheers,
Robin.