what kind of fish will you use? since you will be using mechanical filtration anyways, you could just start them in the 150gal tank in the greenroom, or even the basement?

If you have 4 barrels for growbeds, that wwould be about 200 gallons of media, roughly half to "flood" the barrels, so 100 gallons to flood, that would just about empty your fish tank, so you'll probably want a sump, and make a "chift pist" system...."Constant Height In Fish Tank, Pump In Sump Tank"

if you are raising a cold water fish, you should check the night time temps of your green room and may be able to winter the fish in there with minimal heating (like bluegill or yellow perch), but tilapia would need to stay warmer

you want your pump to be able to pump the volume of your fish tank at least once an hour, but more is better (and you may want o expand at some point ).. you can always divert flow back into the tank to add aeration. and you want it to be able to pump that volume at the "head" you need, that is, how high from the surface of the water you need to pump, you can get a submersible pond pump that would handle the volume you're looking at at a big box store like home depot or loews for under a 100, use pvc to plumb it - oversize is better, like 1" supply