How easy/straightforward/time-consuming is asking CC to put it in bridge mode? Also once done, is there anything specific of which I need to take care of before setting up Switch and then mesh routers?
Very easy, although depends on local technician. Should take less than 15 mins once the technician arrives at home. Some routers allow access to admin dashboard remotely (can't remember that feature name, it is mentioned in this thread for sure) but one guy will visit and other guy will make changes in that case).
They will, depending on router, set it to specific port. Remember that port!
If you have static IP, remember to note down the IP, gateway and DNS, VLAN ID, Priority for future reference (so that you can configure same on your router).
Which mesh routers are you using? At least one of them has to be set up as a router, right? You can connect rest to it I guess
I use 1x Asus AC68U and 2x AC59U. The two 59Us were earlier connected to LAN ports of AC68U. (before I 'switch' ed)