It has been eluded to earlier in this thread, but the main issue with AM radio in automobiles that have large inverters is RFI (radio frequency interface, in this case a subset of electromagnetic interference resonating down in the audio range and low radio bandwidths). From my [rail transit] point of view, it is an easy fix: just slap a cheap Faraday cage around the inverter. In automotive terms, that Faraday cage eats up a little bit of very precious space, but moreover, it makes maintenance access to the inverter system very complicated.
I'm not as familiar with the FCC interference rules as I ought to have been. But in a nutshell, the electronics industry handles the two main sections of RFI/EFI differently. Ergo, (A) they [try] to avoid interference with other people's stuff by mitigating the effective distance that RFI/EFI can have an impact; and (B) they [must] avoid interference with their own stuff. One way to avoid interference with your own stuff is to eliminate the "gizmos" that are most prone to suffering from RFI/EFI.
I occasionally use the AM band in my search of old-timey Bluegrass music. I can immediately tell which truckers have a huge inverter on their big rig, because I lose the AM station to major league interference while I am in close proximity to the cab. None of these appear to be battery-powered trucks, so I assume that they've got a bunch of electronics hooked in (such as satellite TV/Internet and other data services). Additionally, some manufacturers now provide large battery backups for HVAC systems (Freightliner claims a 10-hour HVAC backup).