Is it currently possible to run nyxt with clisp?
For example, luakit is able to run on luajit.
Potentially, some libraries assume the SBCL compiler. We assume no compiler. Therefore it would depend on our dependencies. Your best bet would be to just give it a try 
Would it be better if Nyxt project uses clisp, instead of sbcl, for internal development and testing as well?
I don’t believe so :-O. We don’t assume SBCL, and Nyxt has ran on CCL, but the best experience is definitely with SBCL. If you do spot issues on other implementations, please do let us know!
It won’t be better to use CLISP for development, as it seems to be half-abandoned. We better target the actively maintained implementations, like SBCL, CCL, ECL, ABCL etc. SBCL is a de-facto standard, and it’s lightning fast at runtime, so it’s better to target by default.
As an alternative development-only implementation, CCL is good with type checking. That’s why we used to have it as a second implementation to test Nyxt on.