Your best bet is to look for his service record or where he lived before WWI and after. Most Units were local National Guard Units that were Federalized for National Army Service. Do you know where he lived before enlisted and where enlisted? Lived after discharge? Do you know when born and died and if he has a Veteran's marker?
Looking at WWI sources I have used in the past I do not find a 44th BTC or RTA or Quartermaster unit. The "W" could be for Wagoneer. The Military was still transitioning from horses/wagons to mechanical transport and the names hadn't caught up yet.
It would be better to post to U.S. Army message board as well as WWI. Also post to a Panama message board. Also look for it's history.