What's involved in getting a cirt for your own design? What would stop
me from basing a design on the C-172? How different must it be to escape
copyright infringment?

You are pretty safe, as far as intellectual property rights are concerned.
What you want to avoid is an inspector looking at your finished product and
telling you that he is looking at an uncertified (or is that
"uncertificated") C-172. This doesn't seem to happen with copies of vintage
designs, but I fear that you would run the risk with more modern designs, if
anyone would dare call a C-172 a modern design.

Some builders have gone this route and there are many kits and plans that
are closely approximate certified designs. Copying a design and validating
your engineering decisions could multiply your build time by decades.