If you were not allowed in the right lane the sign would have said, automobiles only-right lane.
This is an 11yr old thread so he's probably chose a lane by now.
Of course, it's equally true that if they wanted you in the right lane, they wouldn't put up a sign saying trucks in the left lane.
If a ticket would stick is a different question but working in the industry, there is no question the intent is for trucks (and not only commercial trucks) to stay in the left lane. Usually, it is because the shoulder pavement is thinner and they don't want heavy trucks tearing it up, though if you are in it to take the next exit, no one is likely to hassle you over.
up here the "no trucks left lane" and other stuff refers to commercial vehicles. if there was ever a " trucks use left lane", it would normally be accompanied by a sign underneath explain why restricted height ahead, or something like that.
the no truck left lane is to prevent accidents and high way closures in the winter as they can't go up the hill fast and they spin out at the sight of snow and block the highway so in the blocked highway that ensures a lane is open for normal traffic and emergency vehicles. also down hill in some areas they are speed restricted so it keep traffic flowing good.
Re-read the original post, it's the opposite (trucks must be in left lane). It gets used in construction zones when two lanes are maintained, the right lane and a second on the paved shoulder.
Unless it specifies "commercial" it applies to all trucks and the reasons you list apply equally to most large RVs.
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.
