Are you saying that the 308's need an avr and the 305's don't? Besides if I didn't need an AVR I'd put the money towards better speakers.
Neither needs an AVR.
The problem is, increasing your budget for a pro audio monitor won't get you a better monitor without spending at least two or three hundred dollars! These JBL's are a bit of a legend, and with good reason.
Shady offers some good options on bass management, but I would advise you to hold off as you might not need the additional crossover control. With the 305, try setting the subwoofer's built-in crossover at 50 Hz. Any speaker has a natural roll-off and the 305 would best match a 50Hz crossover point on the sub (per SVS's "Merlin" tool).
Since you are playing organ, you can determine which notes are in the crossover realm and just pay attention. If you hear some sort of funkiness going on in that range, then you can always add the extra crossover. However, the subs are designed to be used without external bass management. There would be no need for a crossover in the sub if everyone used AVR's (which control the signals for both sub and speakers).
Also, I have had fine results feeding the RCA or headphone outputs from the TV to the speakers for watching movies. Even used Y-connectors to split out the signal to the sub. Works well. (but if you want remote control of volume you will need to use the headphone output unless your TV allows the RCA out to be controlled by TV remote, which is sometimes an option in the settings menu of TV).