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.

I don't really see how this is much of an IG complaint... If the chain of command kicks back an award they can simply say that they didn't feel the event(s) warranted awarding of a CAB. Your commander being a "coward" has very little to do with the process. Little known fact, ANYONE can put ANYONE in for an award with the proper paperwork. That is how awards work. Your platoon leadership can go about submitting the paperwork. If your Battalion kicks it back then they kick it back. Now, if regards to a timeline, it takes a while. The process can take quite some time depending on how active your leadership is in getting paperwork pushed through. The LTC and CSM are going to be where your longest hold ups are and the S1 shop may drag their feet as well. This kind of depends on how dedicated these people are to doing their jobs quickly, what else may be going on, and the kinds of systems that the unit has in place to ensure the proper processing of paperwork. God help you if the paperwork was lost somehow because that does happen. DA200s are your greatest friend in these processes. Would IG do anything about this? Probably not. Awards are a merit based decision placed on the signatory officer(s) shoulders; if anyone in the chain thinks that they aren't deserved then they are within their rights to shoot them down. There are any number of Soldiers running around right now who were involved in events that fall under the qualifications of awarding CABs that have never, and probably will never be awarded them. It happens.