If it were the belt tensioner, wouldn't there be some noise coming from near the front of the engine? I can't detect any, but maybe that doesn't mean anything. In searching previous posts, that seems to be what other people experience (or shredding belts).
The motor mounts crossed my mind, although they were replaced (with shocks) five years ago ... I don't feel any particular vibration on shutdown like I did when the mms were bad. It might need a new transmission mount though ... I have no record of that being done, and it looked a bit "dry" when I was under there (no huge cracks or deformation that I could see, though).
I did notice the new exhaust hangers hold it a bit snug up against the bumper ... like too snug, and it makes a slight squeaky noise. Maybe that's the issue.
Okyourabeast, I think you're most definitely right.

On the bright side, when I turn up the bass, I don't notice my car's little contribution to the low-frequency output.

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1987 300D, arctic white/palomino--314,000 miles
1978 240D 4-speed, Euro Delivery, light ivory/bamboo--370,000 miles
2005 Jeep Liberty CRD Limited, light khaki/slate--140,000 miles
2018 Chevy Cruze diesel, 6-speed manual, satin steel metallic/kalahari--19,000 miles
1982 Peugeot 505 diesel, 4-speed manual, blue/blue, 130,000 miles
1995 S320, black/parchment--34,000 miles (Dad's car)