Is it bad to leave a charger hooked up to the battery permanently. Mine
is a 12/2 with automatic operation.

Only if the charger isn't set properly.

What we're doing is called "float" service, where the charger voltage is set
lower than it would be for normal "recharge a dead battery" sorts of
applications. You have to check your particular battery's spec, but most
sit around 13.8-ish instead of the normal charger output of about 14.4.