Short answer: no damage could be done
Long answer: it's all 12VDC so no damage could be done. Since your solenoid is from a keyed source, I'm assuming that this keyed source is not powered while in the starting position (similar to how the heat/AC blower does not run while cranking).
Even if you were on the factory trailer wiring 7pin. The charger on the TC cannot pass enough current through the wiring (the truck batteries will carry most of the load) during starting. The circuit breaker on your firewall would open if there was too much of a load on the wiring anyways.
You're fine!
