Lose the battlemap entirely . Use a whiteboard & be more generalized with things like distances & size once the grid is gone. If you're fighting in a huge warehouse district with streets around most buildings. You don't need to illustrate every single building/street.
Don't try for tactical, go for things like "pretty far down the street a car makes a sudden screeching turn pulls down one one of the side streets". The non-exact nature allows the mental theater you mentioned to color in some of the details as the combat evolves. If something happens like a building getting blown up to make some difficult terrain, you have difficult terrain anyplace that seems reasonable instead of "these three squares". Let one of your players manage the whiteboard so you can be free to focus on describing things.
With the grid out the window, a whiteboard in its place, & a player manning the dry erase markers/eraser, the mental theater can spring to life & with the exact dimensions lost to general terms the players stop thinking of movement like chess in favor of describing what they want to do to really add some fire to that theater. If I'm facing off against a rocket launcher with a range of X feet while in my tshirt with a stick, I'm not going to stand at X+1 feet & neither should your players going fuzzy with distance will make it so they stop trying and go with things that play out better in the mind for what they mean from the player's perspective than what they can see from their external godlike view of the situation as a player.