Looks like I got it working. I think my mistake was having
CXX14FLAGS = -O3 -march=native -mtune=native
instead of
CXX14FLAGS += -O3 -march=native -mtune=native
My new Makevars, which works is:
CXX14 = icc
CXX14FLAGS = -DSTAN_THREADS
CXX14FLAGS += -O3 -march=native -mtune=native
CXX14FLAGS += -fPIC
Thanks again for all the help!