Adding 12V Battery in Parallel and Solar Charging

Yes, that would help balance the parallel batteries better. Also try to get the same manufacturer and model battery as you got with your new trailer.

And make sure there is an appropriately sized fuse or circuit breaker in the +12v line to the solar charge controller, as close to the battery as you can. Appropriate being at least 1.56 times the short circuit current (Isc) of your solar module (should be on the module label) but not more than the max fuse size on the label. Why 1.56 you might ask? That's 1.25 x 1.25, the first 25% being the normal fuse oversize requirement to avoid nuisance trips, the second being to account for high solar output due to edge of cloud reflection. Anyhow that is the code requirement for solar.