No, once you start a battle, the region is essentially locked down.
From the rulebook: IMPORTANT BATTLE REGION RULES • A Region with a Battle token is known as a Battle Region. • No further Units from any player may enter any Region with a Battle token. • Place one of the Attacker’s Units on top of the Battle token to remember who the Attacker is. • Structures cannot be Built in a Battle Region. • Structures cannot be activated in a Battle Region unless explicitly noted.
Unless the structure explicitly states it may affect a battle, it cannot. In this case, they do not state they can affect a battle region.
Thanks for the quick reply on this one Grant. I think this is an important clarification to include in the FAQ since as the rules are written it is clear that buildings cannot be built or activated in battle regions but they do not state that the effects of a building activation (in this case tokens) cannot affect a battle region unless explicitly stated.