We always assume we are a truck. I don't know if this is true in other states but many overpasses in Michigan are curved. The sides are lower than the middle and if they are pushing you to the shoulder because of the construction there is a very good possibility you are going to be under the lowest part of the overpass. We would much rather keep our air-conditioning unit on our rig. Additionally, they will almost always push trucks to a lane that is level and off the rumble strips.
I used to be a traffic engineer for the state of Michigan. The posted height clearance sign is the lowest clearance for any drivable area (including shoulders). General rule during construction was if you couldn't accommodate non-oversize trucks, you don't open the lane to traffic. Usually not an issue because milling pavement or otherwise tearing it up would lower the driving surface increasing the clearance.
Honestly, your aircon was not a big concern. It's not going to damage the bridge. A big track mounted backhoe hitting the bridge at 70mph is a big deal. I've dealt with a few of these type situations (signed properly but the backhoe arm and dump truck bucket wasn't all the way down). When they tear thru a bridge beam or two, everything stops and it's an emergency. It's also several million dollar bill for the truck's insurance company.
