Should I only run my bilge pump when the engine is off...or can it run
safely when the boat is under power and running?


Your bilge pump should be capable of coming on at any time it is needed.
Running it underway is no different than running any other accessory.

Consider a float swtich to turn your pump on and off when needed, and wiring
the pump directly to the terminals of your house battery. If everything else is
swtiched "off" and you begin taking on water, the bilge pump will try to keep
up with the flooding until it runs out of juice.
Hopefully, someoby will notice the problem before it's too late.