If I go with the BSS processor + my own amps route over the Crown amps that are usually recommended with these speakers, will the audio performance be any worse in any way considering they were designed to be used with the Crown amps?
Theoretically, if you used good amps like the Hypex amps, you could surpass the crown performance.
However, you could also do worse if you don't get appropriately sized amps. I would recommend looking at what crown amps are recommended, and at least try to match the power output rating.
Alternatively, while I'm not familiar with whether this is possible, the JBL AVRs may have EQ for the 705/708 built in. If they support it, you could use one of them instead of your current processor.
Are there any fanless or very quiet subwoofer amps that have at least 1200 watts or more per channel that you could use with the JBL SSW-2 subwoofer?
With regards to subwoofer amps, if you want big power, then you're going to need a fan. I'd recommend getting one of the crown amps for that, and if noise is a big issue then maybe replace the fans with quieter (Noctua makes some good ones) fans, if you are comfortable taking it apart.
Alternatively, Hypex makes high power amps, and others also do, just ensure you have good ventilation.