A few years ago I was in this same position. Our vet had a groomer that they rented space to and she gave Mabel her first buzz cut. Mabel didn't freak out at all, although she did get annoyed by the end. But I think she felt a whole lot better afterwards and it showed in her play! She would run and jump and have a ball. Those little mattes start pulling and start to hurt - so address them as soon as you can.
Since then, the Mabel buzz became a regular thing and she sits while we do it - we invested in our own razor - got one at Target. Our other cat however, freaks out at the sounds of a razor, and she's never been shaved except for her spay surgery.
Good luck!