Considering your alternator (assuming OEM) will never output 150A for more than a few minutes anyway, As dwh stated, 1/0 is beyond adequate. as welding cable has some higher ampacity over typical building wire, & were you needing to squeeze numbers, You could easily go as small as nr.2.
With the distance between batteries, you should use 2 circuitbreakers, one at each battery connection.
Often its fairly safe to omit this one. Small systems anyway.
Your situation, "310watt panel" is most likely a nominal 24-36 volts, so at its absolute maximum output, That module cant produce enough energy to damage a 10awg conductor. ("module" btw, is more correct term... Is my pocket protector showing ?? )
Further, A fuse correctly placed in this circuit, it would be adjacent to the module where its not readily accessible.
YES ! Place a fuse or your favourite OCPD in this circuit. As close to the battery as practical.
Your solar controller probably suggests a size for this one. If not, & assuming 10awg conductors, 30amp max.