To be clear, Skyview signals used to drive servos use pulse width modulation (PWM). It isn't a simple turn the motor on, turn the motor off situation. Auto-trim changes the pulses to slow down or speed up trim response. If that will work for you really depends on how your relays respond to changes in PWM, if at all. Since relays aren't designed to use pulses, they may not even work well for normal trim, much less auto trim. And if these are mechanical relays we're talking about, they're probably going to wear out much faster by receiving PWM inputs. You can experiment if you want. You may get lucky. But if I were you, I'd just ditch the relays and get TRIMAMPs (you need one for each servo)
Some terminology here - I think by "servos" you're referring to the trim motors, which are almost always just 12V DC motors - not "servos".
I also think you're conflating two functions here - first, the "Auto-Trim", which merely trims out the force on the A/P servo, _IF_ you enable the capability in setup (I do not, as I fly IFR and think it's asking for trouble in icing conditions, per Colgan, but for VFR only, autotrim is fine).
Secondly, the "Trim Motor Speed" control for varying trim motor speeds dependent upon Airspeed (See pages 18-7 through 18-10 of version AM of the installation manual), which is what's controlled by PWM. One can set all speeds to 100% and not make them speed dependent, in which case there will be no pulses and the full voltage will always be delivered to the trim motor.
"Auto-Trim" is independent of "Trim Motor Speed" control - it just activates the trim system automatically to trim out servo force when the A/P servos are engaged. If manual trim works correctly (push the buttons at whatever IAS) then Auto Trim will work as well.
Now, if this were my plane, and my trim motor pulled < 2A, I'd get rid of the relay board and just use the A/P controller to run the trim. But if it does pull more than 2A and you can't get a TRIMAMP yet, just use the relay deck to run the trim motors until the Trimamp is available, then replace the relay deck with the A/P controller and Trimamp.
For the OP, you cannot use the A/P controller to control a relay board. Pins 3-6 and 10-13 are INPUTS to the A/P controller from the trim switches - they're not outputs to the motor. The motor outputs are pins 7-8 and 14-15, and switch polarity to determine motor direction, which doesn't work when running a relay deck - the relay deck uses the same types of grounded input pins as the A/P Controller. The A/P controller replaces the relay deck, but can only run a 2A motor, hence the need for the trimamp.
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) draw. None of the RA trim servos draw more than 1A, which is why the A/P controller will work fine without the TrimAmp, and without the current relay board that you have.