If I remember correctly, there is a mute LED on the circuit board that follows the position of the mute button. Depending on your circuit design ability, you should be able to use this to drive the logic inputs of a Henry Engineering SuperRelay, or a similar home-brew version of the same circuit. One side of the LED probably goes to +12/15 volts via a dropping resistor and the other side goes to ground via a collector on a transistor. A voltmeter should be able to tell you what you have on the LED, along with looking at the general layout of the traces on the circuit board.
Provided the LED doesn't do something silly like blink (in the solo position), you should be able to do this. Personally, I would determine which side of the LED goes low when the mute button is engaged, then run this wire to a 1/8 mini headphone jack. Drill a small hole and mount the jack on the back of mixer. Space is generally pretty tight in those mixers, but it can be done. The Henry box has both an inverted logic input and an non-inverted input, so either way, with a little experimentation, you'll figure out which way turns on the relays when you want. If the board does have a solo position that makes the LED blink, you might want to modify the circuit so it is disabled.
The SuperRelay also has a handy 120 volt outlet on it that allows you to hang a blinking ON-AIR or RECORDING tally light outside the door.
If you want a circuit that can turn on and off a relay from this logic signal, it should be pretty easy to build. You can then control whatever size relay you want. I would take a transistor, like a 2N3904 to drive several 2 Amp DIP relays, then use the contacts on the relays to perform the speaker muting. I would be very careful building a tally light relay part, if you do build this. Make sure the circuit is fused and can't cross paths with any other signals in the box. Keep in mind that your circuit is not UL listed, so if it is built for a client, the safety of the circuit (and the liabilities of its use/misuse) are your responsibility. If they decide to run a toaster on the outlet and burn down the building, it is still your responsibility.