Bummer! I went through a learning curve when lowering my CX 5 with coilovers. Lowering is lowering, same interference issues regardless.
First, I would double check the springs are in the mounts correctly. The coil fits into a molded fitting at the bottom. If it's not in this fitting it will make noise. I installed a spring incorrectly and it told me so....
Upper and lower shock mounts will make this kind of noise if not torqued correctly as well.
Sway bar links not torqued correctly will clunk as too.
Did you use stock or get adjustable links on the rear. I broke a stock link when I put the rear a Megan rear sway bar on the tightest setting. I bought a set of adjustable BW links. They are have steel sockets vs plastic OEM. Latter I lowered and the links were hitting the frame. I had to shorten the links to avoid them clunking. Look for fresh scratched paint on the links and or frame. Best not to jack up the car to see the interference.
These seem to be sturdy end links. I'll know if 10k mi or so. I have these front and rear so the sway bar is parallel to the ground with the car on the ground. Lowering will change the geometry.
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Can you move the upper control arm with your hand or stick a pry bar on it to see if it has a defective bushing or not tightened to spec. These are difficult to get to and may need to be tightened again. The installer should be willing to put a wrench on every bolt if you don't own tools, while you stand by and watch...
Here is an adjustable lower control arm I'm using. Easier to install and adjust. If I make it easier for the tech, I'm more likely to get a better alignment. Install was supper easy. Not what many, if any folks on this forum use.
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FWIW, I ended up learning how to do my own installs after going through stuff like the above and much worse. I could write a book on the stuff I paid to be installed and had to fix. So I slowly bought tools and learned...
The final straw, I paid a well respected shop to do an engine swap for me. Latter I found a fuel line crushed under the intake manifold. I was fortunate to find it before it burst and burned my car down.
I'm curious if your front end camber is in spec? Mine was out a couple degrees after lowering it 2" in the front, 2.5" rear. I bought camber bolts for the front to fix this issue.
GOOD LUCK