I've never had a pet in carrier weighed on an airline, so I can't imagine that Amtrak would try it. The closest I've gotten while flying was a question at the gate: "Does that dog weigh under 20 lbs? Yes. OK Thanks."
I'm not even sure where they would do this at South Station and definitely no way to easily do it at Back Bay.
As long as you don't look like you're buckling under the weight of carrying the dog, I think nobody will even look twice.