Yes, as long as the solar charge controller is either 1) turned off when connected to shore power or 2) has a "shore power" charge mode, like ours, which just decreases the peak output voltage of the unit to be at or below the shore power charge controller. Otherwise the solar charge controller can raise the charging voltage higher than the shore power charge controller and potentially back feed the shore power charge controller.
Having the solar on while the vehicle is running creates no problems if connected via the trailer plug, it will prevent back feed.
Wait...what?
I'm sorry, but this makes no sense to me.
What solar charge controller do you have? I've don't think I've ever heard of one that has a "shore power charge mode".
One of the purposes of a solar charge controller, is to PREVENT backfeed from the battery to the solar panel when the sun is down (otherwise, the solar panel becomes a heating element and consumes the stored power from the battery).
In other words, no worries about "backfeed" IN to the solar charge controller from the battery side.
The solar charge controller doesn't accept a backfeed.
As for a solar charge controller feeding power into the 12v bus at a higher voltage than the mains powered battery charger and overpowering (backfeeding) the mains powered battery charger - where is the power going to backfeed TO?
The electrical grid? That won't happen.
So what if the solar charge controller raises the voltage on the 12v bus? It won't matter. What if the battery was at 14.4v and the solar charge controller is disconnected and then you hook up a mains powered charger to the battery? Wouldn't the BATTERY backfeed the mains charger - even without the solar charge controller connected?
No, because the mains powered charger doesn't accept a backfeed. If the voltage on the 12v side is higher than what the mains powered charger can overcome, then it won't be pushing any power into the 12v bus, but it won't be accepting a backfeed and converting it into 120v either.
A device designed to hook up to a battery and charge it, isn't going to allow itself to be overpowered from the battery side. At least, it shouldn't. Not by any voltage in the normal "12v nominal" range anyway.
And a trailer plug has got nothing to do with anything, except for hopefully being controlled by a relay so it's disconnected when the key is off, so the engine battery doesn't get drained along with the aux battery.