If you are going to have a battery trailer, I think it has to have it's own cooling/heating.
I don't think so, especially with cooling. A battery trailer would only need to provide a constant-ish, low-ish level of power to the main battery. Active louver air cooling would be sufficient.
In any case, a battery trailer doesn't make sense. Its a complicated, expensive, and low volume solution to a problem that will be eclipsed by technology advancements in barely a few years.