Often times "dealer installed" does not include any electrical wiring that might be necessary. I did a lot of research before buying the lift that I did and I found that the installers that I talked to requested that the wiring be completed prior to them coming to install the lift. That way they can properly test it when they install the lift. I ultimately installed my lift myself, and I'm glad I did.
Based on the motor nameplate, the only safe, reliable, code compliant, solution is to run a new dedicated 10awg 30A circuit for the lift. Do you have an attic above your ceiling?
You say your breaker box is full, and that your string of 8 outlets are only for intermittent use... OK, remove the breaker feeding those outlets, and move the wire to another circuit that feeds other outlets in the area. Just don't land 2 wires on 1 breaker unless the breaker terminal is rated for that. Now that you've created an open slot, use that for your 30A single pole breaker.
Did the lift come with that plug on the end of the motor? Shame on those China lifts if it did. Just because the wires inside the motor, or any internal wiring on the lift might be 14awg, you still need to connect the lift to 10awg wiring per code. My 10,000lb Mohawk came with a 3 prong 30A twistlock prewired to the motor with 12awg SO cord and the mating receptacle was supplied in the parts bag. It required a 30A breaker (mine was 30A at 240V though) and thus from the mating receptacle to the breaker box had to be 10awg.
Brian