Burst Bomb first, and the grenade throw animation gets canceled by the flick. There isn't really any strict timing because it can only happen at a certain speed, you can just tap R, ZR quickly.
The range where the animation cancel is useful is actually quite small, but it becomes larger with Damage Ups and it's still rather important.
I use the Carbon with my Damage and Bomb Range build. I am generally pestering opponents with Burst Bombs cross-map, and if I hit someone at a close-ish range I can react with an animation canceled flick for the kill.
In normal CQC fights, you simply need to have a perfect understanding of your spacing to know if you are close enough to just flick, or are far enough to have to animation cancel. If you make the wrong decision and die, it's simply your fault you died and you're juat gonna have to gain more knowledge and experience in those CQC situations.
For gear, I feel that Sun Visor, Squidmark Sweat, and Blue Slip-ons are the best naturally rolling gear you can get for the Carbon.
If you save scum (check the guide in my sig) you can go for a damage main hat (I have the Snorkel Mask) and the Squidmark Sweat with a combination of Damage and Bomb Range rolled between them, as that will manipulate the RNG in your favor. I wouldn't do that at first though unless you try out my above recommend build and think you need more Damage Ups than you can get on your Sun Visor.
Good luck and have fun!
*Edit*
A few things I forgot:
You have to let go of LZ for it to work.
You can press R, then mash ZR. Since the cancel can only happen so fast, there's really no reason no to mash or at least double tap ZR to guarantee you get the timing.
