How can I install a recessed light container in an abnormally thick ceiling?
There is a category of power tool called a multi-tool that is basically a very small saw usefull in cutting in tight spaces.



After you cut the initial hole, this tool could be used to trim away  the area in the uper section on an angle to make room for your retaining arms.

The tools come in corded and cordless version, and at various price points. The are very handy and can do other things, such as spot sanding and grinding.