We have read dozens of documents now - some that contradict each other - and cannot find a solid source of truth. Does FIPS 140-2 compliance allow for the use of elliptic curve cryptography as a key exchange for TLS communication? ECDH and ECDHE - if so, which P-xxx key sizes?
If you look at the FIPS Implementation Guidelines for FIPS-140-2 here https://csrc.nist.gov/csrc/media/projects/cryptographic-module-validation-program/documents/fips140-2/fips1402ig.pdf
In section D.8 Key Agreements, you will find the recommendations. This recommendation points to NIST SP-800-56A, where in Appendix D, there is a table of "Approved ECC Curves and FFC Safe-prime Groups."
SP-800-56A has references to FIPS186-4 and RFC4492 (for TLS) and RFC 5903 (for IKE). These references should be enough to justify use of ECDH in a FIPS140-2 approved device.