CURT 56310 for my CX5: powering option

Hi,
Someone gifted me with CURT 56310 and I quickly installed it and call me a dumbo - didn't read the instruction manual fully. Came to realize that I need to run a wire to the battery! Its cold and windy and hate to jack up the car and route the power (black) cord fiddling with that grommet and then engine bay and working underneath the car.

What if I instead connect the power to the ubiquitous little 12V power inlet I have in the trunk instead? For example, I have https://www.amazon.com/dp/ (commissions earned)lying around, will it work? It'll save me a ton of trouble. The install kit does come with a red in-line fuse with instructions so was thinking might be safe?
Alternatively, I have my son's (outgrown) https://www.amazon.com/dp/ (commissions earned)y with an adapter having 2 leads. Can't I use that instead to power the harness?

I'll need the harness working very soon as I need to tow a substantial number of times next week *moving to new location*. My towing activities is very, very limited and most probably the next time I'll need to tow is perhaps 2 to 3 years from now. Thus my need of battery or connection to the trunk 12V port is very temporary.

I'd use the power socket over the battery.
/Edit I'd not if... Lol

When I installed my towing harness I installed a 15A rated ,fused cigarette lighter adapter on the end of the long power wire(similar to the Carviya one advertised at the top of this page for 7.99 $ at the big "A") to use the 12v socket by the rear hatch.I didn't cut the long wire in case I need to run it up front later.I haven't towed yet due to a change in plans but everything works(turn signals,brakes) according to the little harness tester I bought at the awesome Harbor Freight.I put the connector in a baggie tied off with a rubber band and stow the coiled wire in the spare tire well.Keeps things neat.

I did it this way because it was easier and my towing would only be 1 or 2 days moving household goods.Sounds like your situation is similar. Jmaz

Thanks 7engine and Jmaz: This is perfect. I just bought https://www.amazon.com/dp/ (commissions earned). Now, I only have to figure out the best way to connect to these ring connectors the stripped black wires so that the connection's rock solid and -ve and +ve don't end up touching each other.

I used a t-tap to tap into the 12V outlet in the cargo area.