I had the old Battleground games. I thought the Battleground AI was weaker than the Campaign Series AI. Like Campaign Series, the only way to improve the AI was to adjust where the AI fell in the combat tables. I'm not sure if the adjusting the AI threw in additional troops for the AI. I don't believe you can adjust the AI's tactical abilities. I think I read somewhere on one of these threads that the larger the scenario, the more trouble the AI has. I don't have the Matrix version, so if my memory fails me, or if Matrix has made some improvements, someone else will have to speak up.
Regarding PBEM play, I never tried that. My concern was the Battleground games played through a series of phases within a single turn, which would require the swapping of files several times before completing a single turn. It apparently works, however. My impression was if you bought the game strictly for play against the AI, it wouldn't be long before you found the AI to be a pushover.
You've seen my PBEM abilities in Campaign Series, so that should give you some pause.