The game keeps randomly stopping me to tell me it's disabling achievements because I accessed the console. But I haven't! This gets my goat quite a bit because I already have and enjoy Doom 3 Vanilla quite a bit and my only interest in BFG Edition is getting the achievements.

I believe the test is whether you have the console enabled (com_allowconsole 1), not whether you actually accessed it. I don't know what in your set-up would set the console to being enabled, but try putting this in your base/autoexec.cfg: