Hey Andrew, it was probably 2nd battalion. I did some googling on WW II infantry regiment organization and found that Companies A-D made up 1st battalion, Companies E-H made up the 2nd, and Companies I-L the 3rd. I also browsed through your excellent profile. I think you have a great deal of information for piecing together the circumstances (location, type of action, etc) of your great uncle's death. If you know his platoon or the name of his lieutenant, you might be able to narrow it down further. I doubt that you will get any more specific answer than that, unless you are lucky enough to find an eyewitness account that mentions him by name.
The other town prominent in the reports, Flamierge, is near the left-hand side of the oval on the map, where the roads form a sharp angle. A slightly tighter zoom on googlemaps brings out the name.