I'm not completely sure what you are refereing to as 'church date', but if you mean simply attending or participating in a church related function on a friday night rather than go to the movies with your date, then... It wouldn't be awkward at all, not would it test your spirituality. Where is a better place to spend time with someone you care about than before the Lord. Not to mention, what better way to fellowship and rejoice than with someone that you are that in touch with. Sharing yourself spiritually with your companion is possibly a stronger bond or stronger connection than any other that I can think of. Power to everyone that can fellowship in any atmosphere, with everyone that they can.
*God Bless*
If we both are Christian, I don't see how it would be uncomfortable to attend a church function together. If one of the couple were not a Christian, it might be awkward, but as bwren said, what better way to spend time than before the Lord? Perhaps even fellowship your date into the church ? As far as testing the "spiritual level" of your date, I don't know that attending church functions would do that, but deep discussions of a spiritual nature one-on-one I think would be better than meeting in a crowd.
Roz