Do you know what voltage and current the solar trickle charger provides? What are the specifications and maybe you need to measure its output voltage in your present location. It need to be about 14vdc (needs to be about the same volts as the engine "generator", not much higher or lower, and produce about 1 amp to keep the battery charged. That would be 14 volts x 1 amp = 14 watts.
Less than 14 volts dc will simply not charge the battery, maybe could discharge it if below 13vdc.
I ran 6 gauge battery cables to the posts of the winch relay under the front hood. My battery tender has a quick disconnect cable connected to my winch relay posts, that is my "bus bar" extension of the battery in the front.
You could possibly rig up a connection point in the front if you have a winch or fuse box that is always on from the battery.
Could also rig a quick connect for a portable lithium battery starter if the battery is too weak to start. Are you also sure the 12vdc accessory plug is live +12vdc with the key off and engine off (I presume you already did this, but I need to ask to cover all of the possibilities).
If you had a good voltmeter and check everything with it, it could be very educational.
My 2020 P500 can sit for 4 weeks with no trickle charger in an outdoor garage, and never have a problem starting it, so I think you could have some deeper problem. For note, when the engine is running, you should have 14.1 volts or so in the whole electrical system, so shifting should not be a problem. However if you have a loose connection at the battery, or the negative cable to ground, the shifting could be at only the battery voltage or intermittent voltage with the engine running.
Hope the best for you, that is difficult when you cannot get 110vac power to you storage location.