How can we run all tests using C++ specified compiler?

[Jeffrey Walton]

I'm testing different configurations of a C++ library. When I test on
AIX with XLC I am catching an error:

-bash-4.4$ OBJECT_MODE=64 CXX=xlC ./configure
checking for a BSD-compatible install... ./install-sh -c
checking whether build environment is sane... yes
checking for a thread-safe mkdir -p... ./install-sh -c -d
checking for gawk... gawk
checking whether make sets $(MAKE)... yes
checking whether make supports nested variables... yes
checking for style of include used by make... GNU
checking for gcc... gcc
checking whether the C compiler works... no
configure: error: in `/home/noloader/cryptopp':
configure: error: C compiler cannot create executables
See `config.log' for more details
...

The problem is using the wrong compiler:

configure:3381: gcc    conftest.c  >&5
Assembler:
/tmp/cczCkD4X.s: line 9: Only .llong should be used for relocatable expressions.
configure:3385: $? = 1
configure:3423: result: no

Our configure.ac has AC_PROG_CXX after the various init's, and lacks
AC_PROG_CC. We don't do anything with the C compiler because we are a
C++ library.

How do we tell Autoconf to use the specified C++ compiler for all tests?

Thanks in advance.