12V trigger inputs on integrated amps, when found, are usually on higher-end units with MSRPs 250% to 500% of your budget, like Audiolab 8300A (~ $1200) Anthem 225 ($1999) and Parasound Halo Integrated ($2495), and although all can drive 4 ohm loads, none of them have A/B speaker switching.
Your best bet is to probably find an amp with A/B switching that can handle 4 ohm loads (2 - 8 ohm speakers in parallel) and get an external device, either a 12V trigger actuated outlet like Xantech AC1 or some kind of auto-sensing power strip that senses current when the control component is turned on and then powers on other receptacles.
Steve