There was a bug (17322941) filed for the FIPS issue you linked to. It was resolved with ODP.NET 220.127.116.11, which you can download as part of the Oracle 18.104.22.168 patchset or as part of ODAC 12c R3 when it is production.
The fix requires upgrading both the client and server side. On the server side, you must use 22.214.171.124 or higher.
Update on August, 2015:
Oracle DB 126.96.36.199 can be patched to support managed ODP.NET with FIPS. Refer to Bug 17551699 for minimum patch set update/bundle patch. Or you can just use the latest patch on top of DB 188.8.131.52 to be supported.
Thanks Alex. I'm using 184.108.40.206 on the client side but the service side is 11g (220.127.116.11.0 according to v$version) which would explain why we're still seeing the issue.
Gee Alex been waiting for a FIPS140 fix for LONG LONG time..
too bad will not work with 11g20x.
So I will have to wait for Oracle 12c to come our way on server side.
But fortunately, my section is usually the guinea pig for new Oracle installs...
and there is always Christmas.