How long (roughly) does it take for a CAB request to be approved?

While it sucks to not be awarded in a timely manner, 3 weeks isn't long at all. I didn't receive my CAB until 2 months from redeployment. I had been in a firefight with my CSM doing a ride along at one point. The bigger issue is that you have a 1LT or 2LT talking shit about your commander with a private. Had I have been there I would have asked the LT to come talk with me aside for a minute and explained that we can denigrate our superiors in front of subordinates. I think that there's probably some significant problems with your platoon and battery leadership if your PL thinks that he can just tell you that your commander is a pussy.

Length of time from incident to approval varies but 3 weeks isn't uncommon nor is it an exceptionally long wait time considering the recommendation for the badge has to travel from Company to BN, to BDE and then to the 2 Star for approval a lot like an MSM or Bronze Star. But from where I sit you guys have bigger issues than the time it takes for a CIB to process.

In the Army I grew up in, Platoon Leaders don't tell PFCs "that our battery commander is too much of a coward to push for the award that we earned on our deployment" That's highly unprofessional and tells me as an old Soldier a lot about the PL and PSG. Took me under two minutes to type the awards process and progression for you so you could see the steps a CAB has to go through to be approved and that 3 weeks isn't unreasonable.
You could also look at AR 600-8-22 and figure it out, there's a table there that lists the process and procedures (which is what your PL and PSG should be telling you). There's no fight for your commander to make unless the recommendation comes back disapproved.

The IG in my experience seriously looks into any grievance they are presented with, but there is a reason CABs go through an approval process. Make sure when you talk to the IG you include the " I was told by our platoon leader that our battery commander is too much of a coward to push for the award that we earned on our deployment. The platoon sergeant and leader are both mulling over all of us as a platoon taking this matter to IG when we redeploy stateside here shortly if we're denied," That always helps paint a picture for them as to the units leadership climate.
In any case, I appreciate you serving and that you get your CAB if authorized one.

Way to throw your battery commander under the bus, maybe he has a RP account and now knows what you think of him, or HELL be super private use the open policy and walk right and tell him what you think of him since your posting it on social media for him and anyone that knows him what you think. Awards have a process I'm retired and still waiting for DOA to approve or even look at my BSM packet. Takes forever.