Alan,
The meter should be connected into the battery cable going to the controller, and should only come on when the battery switch is turned on. If wired like this, it will be fine to leave it connected to the battery during charging, as long as the battery is switched off.
If you have quick connectors on the meter, it could be left attached to the battery (and the battery turned on during charging) as long as the speed controller is disconnected from the meter. It could then be used to monitor the battery voltage during charging if you wish to do so. With my
LiPo battery packs, I actually plug the meter into the charging circuit so that I can monitor the current and voltage etc. and record how much capacity is being put back into the battery whilst charging, so I can compare this figure to the capacity used while riding.
During use, the meter should remain reasonably cool, as it only consumes a tiny amount of current and therefore does not generate much heat, and I've never been aware of my meter getting even slightly warm.
I'm thinking about modifying one of my meters, so that the shunt section can be mounted near the battery, with three small wires linking it to the display unit on the handlebars. It should then be possible to put a changeover switch on the two positive wires from the shunt, so I can also monitor the regen current simply by flicking a switch.
Alan