Should the Tone Arm be Removed When Installing Cartridge?

You would have to hit the tonearm with a hammer to damage its bearings or more likely drop it on the floor. I have never seen a tonearm damaged in the process of mounting a cartridge. I have seen several bent cantilevers. 

Normantaylor, you could do it either way but here are the considerations. 
You do not have a removable head shell. You have a very bouncy suspended turntable and three bolts and the tonearm board is off.
IMHO it is relatively delicate work made more difficult with a moving target. I would remove the tonearm board with tonearm and lay it on a towel to put a new cartridge in. It is safer this way with almost no additional work. It might take a big two minutes if that to unbolt and bolt on the tonearm board. Do not try to manipulate the cartridge wires with your fingers. Use small needle nose pliers or tweezers. The pliers are much better and you can probably get one at Harbor Freight for $2.00.