When the battery runs dead....

Yes, allow the truck a few minutes to charge the battery some then you can do what you asked. If you have a large Alternator on your truck you can hook up the trailer battery with jumper cables, this will help charge your battery much faster.

Run the trucks rpm above idle. My HP Alternator in our truck will put a large amount of charge back into the battery bank in a short time.

Recharge the batteries for the awning daily, before they get low, in charge, let alone go dead, usually first thing in the morning.

Running the batteries that low will reduce their life so they will reach this point sooner and sooner on future trips. If you know that you are that short on battery capacity install better batteries, more batteries, solar, or generator now so you can have the best experience in the future.

If all is connected and working as its supposed to, yes.
But jumpering TV battery to trailer battery, will work faster.

Probably. Bring jump cables to be certain.