There are some good comments here about compression, and yes, start with the studio comp, but play around with the different types, the Dyna comp is something Eric Johnson uses on his clean electric sound, it might translate to a good character on acoustic.
I'm definitely hearing EQ as well as com in the signal path. A five band EQ would be useful to play around with, although I'm not familiar with the EQ settings that best complement an acoustic guitar. I linked you to a 15 minute video that might assist.
Another factor to consider is the Bagg's saturation. We're talking analog saturation here, so try placing a tape drive block in the signal path. I hear gain boost when the player in your video hits the Baggs pedal. Try placing the tape drive block at or toward the beginning of the signal path, because the boost can push the compressor and you can tame the volume on the compressor block output. You want to turn the tape drive block up to the point where you first hear the first hint of breakup and then back the drive knob off. A/B the tone with the drive in and out and find a sweet spot or a range, and you can experiment with the mix knob (same as wet/dry knob) to get youroptimum tone.
The Axe Fx has just about the best verbs available anywhere, and a slight hint of it could help, maybe program it to kick in with a scene change, same with chorus, which is widely used on acoustic guitar. That's not part of the Baggs pedal, but there's no law saying that you can't produce a preset that sounds "better" than just a Baggs pedal
You can also head over to Axechange and search for acoustic guitar presets and check them out. They may not be completely to your liking, but one of them might have a good EQ block setting, comp setting, etc. You can copy those settings and drop them into your preset, tweaking them further until you get the best sound.
The moral of this story is that you have a fire-breathing tone-creating DRAGON in your hands, you ought to be able to find an ass-kicking acoustic tone within the Axe.