Is it ok to start with a battery charger attached????
YES. but...
Most modern battery "Smart Chargers" which by the way are the only kind I recommend, Even have a special setting for it.
My latest "Automotive" purchase (not counting tires and gas) is a Deltran Battery Dender Delux. It has 3 modes
Default: Battery tender 1-2 amps. fully automatic
Battery boost: 20 amps. runs things up to 14.1 VOlts and then drops down to Default (Tender) NO auto return to boost mode.
JUMP START 70 amps for 1/2 minute max (I think, not sure of the time lmit)
Last one I had. also had a Jump start mode. it wore out.
SOME of the older chargers.. Might have an issue but .. I doubt it.