Can an Army unit force you to utilize a specific mode of travel when traveling somewhere on orders?

Yea. They can pretty much dictate. It's their funds. Probably someone took a look at a DRA and figured there is less risk in you flying in comparison to driving and would rather pay extra for the flight.

If you are on orders to Bragg, whether it be for PCS or TDY, the unit has to account the distance needed to be traveled and the amount of travel days authorized. Per Army regulations, you are "not allowed" to drive more than 400 miles per day. If the total distance is greater than 400 miles, they would have to give you an extra travel day which may not be cost efficient, hence why 90% of CONUS PCS moves give you 2-3 days unless its to the west coast. TDY is almost always more cost efficient to fly the individual to the location.

Yes, Government may specify mode, especially if there is a specific arrival time and there is not an authorization for in and around travel by government auto or POV. The purpose of travel (like a school) may specify NO POVs. What you can do is engage the travel approval authority from the DTS chain and show the lower constructive cost of POV travel Vice commercial air and commercial ground transport. If the answer is no, then it is no. Yes they can do that.