I think it's not enough to provide a different compiler path to cmake, since this is a cross-compiling scenario, so a whole cmake-toolchian is needed.
The emscripten SDK has one under "cmake/Modules/Platform/Emscripten.cmake", but trying assimp via (e.g.):
cmake -DCMAKE_TOOLCHAIN_FILE=~/scratch/emsdk/upstream/emscripten/cmake/Modules/Platform/Emscripten.cmake ..
...creates a Makefile, but then fails to compile (looks like some configuration problem).
/Users/floh/projects/fips-assimp/assimp/code/glTFExporter.cpp:585:35: error: ordered comparison between pointer and zero ('aiVector3D *'
(aka 'aiVector3t<float> *') and 'int')
if(comp_allow && (aim->mNormals > 0)) idx_srcdata_normal = b->byteLength;// Store index of normals array.
...so it looks like it's not done with a simple cross-compile, but needs some code changes too to be compatible with latest clang.
There's an open ticket in the assimp repo about emscripten-cmake support:
Might be worth pinging there.
Cheers,
-Floh.