My reference lists these sailing/arrival dates, which match the ones you submittd:
44-09-05, out of BPE, the GENERAL W.H.GORDON, with part of the 44th Infantry Division, arriving Cherbourg, France, 44-09-15.
44-09-05, BPE,(Unnamed British ship) with part of 44th ID, arriving Cherbourg, 44-09-15.
Other elements may have been aboard, as well as the 44th ID.
I don't see either of the units you list as part of the 44th ID's organization.Instead, I see the 749th TB, and the 772nd TB listed as being attached.
A second reference for the 778th TB does indicate that the battalion departed BPE and arrived Cherbourg per the above schedule.
The 778th TB was credited with these campaigns:
Ardennes-Alsace (Battle of the Bulge), 16 Dec 44 / 24 Jan 45 Central Europe, 22 Mar 45 / 11 May 45 *Normandy, 6 June 44 / 24 July 44 *(THIS HAS TO BE AN ERROR, FOR NORMANDY ENDED WELL BEFORE THE ARRIVAL OF THIS UNIT IN EUROPE). Northern France, 25 July 44 / 14 Sept 44 Rhineland, 15 Sept 44 / 11 May 45.
It was located at Salnau, Czechoslovakia, when the European campaigns ended.
The 44th Infantry Division was only credied with Northern France, Rhineland and Central Europe.
It is late now, butwill try to "match" your units to the battle at Dreibkopf, Germany, tomorrow.
I hope that this matches your profile. Let me know. OK?