What you are describing really sounds like a bad tensioner assembly (as mentioned above), look at the tensioner while the vehicle is making noise to see if it is smooth or jumping around any (like it is moving from loose to tight back to loose back to tight very quickly)… but pretty smooth when not making noise…
If you have a 1/2" drive impact gun, put a socket on it and spin the alt really fast to see if it makes noise, you may have bearings going bad in it… You might have to remove it for access to the pulley nut, or get creative with a universal joint adapter (impact preferred but standard will work, best if it has a few rounds of electrical tape around it to stiffen it up)…
Are you sure the entire mount has to be removed? I’d expect all that’s necessary is just the mount’s “through bolt”, which holds the body part of the mount to the engine part. Removing the front-mount through bolt is a relatively easy job on my Corolla, maybe 15 minutes. A little jacking is needed on the engine beforehand of course.
Still way better than the old 3,4,5 V belt set up where you had to pry the alt to tighten the belt, to tight and you can burn up the bearings, to loose and it will squeal, some of the PS pump belt adjustments were just stupid, not to mention some pulleys ran 2 belts and some only one, and all the belts were different lengths, no thank you, I will move a motor mount for a serp belt with auto tensioner…
More involved than many. I’m not a mechanic, so maybe it’s not all that uncommon. But I was surprised when I looked up the tensioner and saw the cooling connections. Weird.