Free Class A is good. Very good. I just bought some to add another chimney to my house, and it shocked me, having not bought any in over a decade.
I also recommend figuring a way to support the stack from the bottom. If nothing else, some sort of steel plate with a hole cut in it. I only know Metalbest [Selkirk], but they offer a small base piece, just a ring-sort of thing into which the bottom section of pipe will screw. You can bolt that little piece down, if wanted. I've never used it, just seen it in catalogues. You could use that to attach the pipe to a steel support plate of some kind. It would last forever. I would also secure the pipe run at the top of the masonry chimney, something simple would work. Particularly if a stove pipe section or even the cap extends out beyond the masonry. You don't want wind rocking that pipe around.
Also, as you lower the pipe into the masonry chimney, make double sure you have tightly secured the locking bands that go on each connection between sections. I have seen one case in which winds moved the stack very slightly, and over time, somehow, one section of this run had come loose. It had unscrewed, in other words. It did not come apart, the upper sections of pipe were too heavy, but it was quite loose and an unusually high wind or storm could have made life interesting. I doubt there would be any trouble with your Class A protected down inside the masonry, but just be sure.