Batteries test okay, but…cranks slow.

If it cranks slow, and batteries and cables are good, it needs a starter.

I'm curious how the "mechanic" checked them

 

I bet one of those batteries will fail the eyeball test and will be swelled up.

 

Check all the battery cables, make sure they are tight, and corrosion free. If they are corroded, clean them with a wire brush and baking soda and water or any battery corrosion cleaner. Check battery cables for the same at the starter. This is very easy to do, I would start there.

 

This thing works well and is inexpensive. A bad/failing APU alternator could be draining everything as well, how old is the one you have?

 

Earlier International Prostars were famous for having crappy battery cables / unprotected / not much insulation.
Don’t know if the newer models have the same issue.
I had a truck that the starter seemed to drag, found it was the main battery ground on the frame.
Another truck right from the manufacturer: they put masking tape on the frame ground stud before painting, but never removed the tape before the wire was tightened down. Over a couple of years the tape deteriorated leaving the ground wire a little bit loose.

 

A poor/dirty cable connection between any component will seem like a dying battery. Clean off every connection and check for low resistance and good continuity.

 

Could be a bad diode in the truck alternator...
It'll charge just fine but when you turn it off power will back flow thru it..