There is a third option: Live with the hum. You could try switching from speaker wire to a ground wire with spade connectors like the ones offered by BJC:
This may provide a more secure connection that will further reduce the level of ground hum, potentially to an acceptable level. Another thing to try, since there is no dedicated lug, is to move the point where the ground wire is connected on the amp. You could even try one of those bolt-on ground lugs:
Bolt-On Master Ground LugThe importance of a solid ground connection is paramount in any 12 V installation. Our Master Ground Lug accepts multiple ground wires (with a power ring on each), and bolts securely to the vehicle's sheet metal, creating a large and secure electrical contact for reduced...
www.jlaudio.com
No idea how well it works.
If you have to replace the amp, you just need something that can drive 2-channels and has a proper ground lug.
If you decide to replace the AVR, get something with XLR outputs, something like the Denon A1H, Yamaha A6A, Integra DRX-8.4, Pioneer 805, or even a dedicated processor like the AV10, AVM70, or HTP-1. If you went for the Denon, Integra, or Pioneer, there would be no need for additional amplification anyway since those units all have at least 11-channels of amplification. With the processors, you will of course need external amplification for all channels.
FWIW, I've had ground loops with every amplifier I've ever owned (except subwoofers, for whatever reason!), and it's always been solved by either XLR interconnects or dedicated ground wires.