I use the FLOTit 2.0 a little differently.
From the conical fermenter I rack to a keg with FLOTit 2.0.
The keg then goes into a cold crash.
After a few weeks I transfer under CO2 to another keg without FLOTit and then gets carbonated.
Keeps the crud out of the serving keg and now the FLOTit keg is ready for the next beer transfer from the fermenter.
I just transfer into a keg with FLOTit 2.0 installed and then put that keg in the fridge to carbonate. The beer cold crashes while carbonating. I serve from that same keg. Any muck from cold crashing is on the bottom of the keg and not a problem.
Often there is debris in the first couple of pints. I think that is from being inside the filter from filling the keg through the FLOTit.
So far I primed 2 kegs with a Flotit and they poured clear from the first pint, which is nice. That option seems good for me for a lot of beers. I haven’t tried force carbonating with a Flotit yet, but I’ll try it soon. Probably I’ll try the method of just shaking with the co2 entering the gas line and see if agitation is enough to get it working. If it doesn’t work, then I’ll just have to “set it and forget it” like I’m in a ‘90s infomercial.
I use these…love them. Have them in place for all my kegs. When I force carb, I carb at 30 psi and then I will hold the keg horizontally and shake the crap out of it for a couple of minutes. Usually do this 3-4 times. Then it goes into the cooler at 38° F. Never had an issue with one of them falling off.
I use the floating dip tube in all four kegs. The beer will clarify from the top down.
Shaking to carbonate really depends on the style. I would leave that lager under pressure and not shake, but that NEIPA would get a shake. My “shake” process is more of a rocking motion to get the splash that creates more surface area that creates more absorption that creates yummy. 
I use floating dip tubes and burst carbonate by laying the keg on its side. I carbonate through the “in” post, and orient the keg so this post is submerged
Here’s my procedure:
- rack beer to keg, close keg with floating dip tube in place
- connect CO2 to “in” post
- set CO2 pressure to 15psi
- purge headspace by waiting for CO2 ingress to stop, then venting pressure, repeat 3-4 times (this step is super important to enable fast carbonation)
- lay keg on side (on a towel, on the concrete floor)
- set CO2 pressure to 35psi
- roll keg vigorously back and forth for 30-60 seconds. The key thing here is to really agitate the surface.
- pause and listen to the CO2 bubbles - bubble rate and how long it takes for them to slow down and stop. This gives a gauge of how fast CO2 is dissolving into the beer. The faster the bubbles and the longer they keep going, the higher the rate of carbonation
- repeat 7 and 8 until 8 indicates you’re close to the desired level of carbonation. I can’t describe how to gauge this; you’ll get a feel for it. Total time for 6-8 is usually around 5 minutes, maybe 8-10 for highly-carbonated styles
- set up your kegerator pressure as desired (see note below)
- put keg in kegerator and let it chill
- 24 hours later, enjoy cold beer that (if you’re lucky/good) is perfectly carbonated. Plus, since you’re serving from the top, it will already be pretty clear.
What you do at step 9) totally depends on your setup. I sometimes leave 35psi on the keg I’m carbonating and shut of CO2 to the other kegs, just relying on their internal pressure for dispensing over the next 24 hours. Other times I turn off CO2 to the new keg and set the regulator back to serving pressure for the rest of the kegs.