Is it possible to configure CLion to compile source files in a project independently?

You could define multiple executables in the CMakeLists.txt for each problem.

instead of

add_executable(projecteuler ${SOURCE_FILES})

you could define

add_executable(problem1 problem1.c)
add_executable(problem2 problem2.c)

Then you get for each executable (problem1, problem2 etc.) a run configuration, which you can run independently. In this case you won’t have to rewrite every time, instead you just add the new source file to a new executable.