If you are a Mac user there actually are possibilities to stream Logic into the Record mixer via Jack or SoundFlower. Check it out
Here.
I've never tried it myself since I migrated to Reason, Record for pretty much all my music production and mixing work. The only projects I use Logic for is TV or Video Games work since R+R still don't support video yet and for jobs like these Symphobia, Vienna Instruments and others are just indispensable.
Another thought:
There is nothing wrong with composing in Logic, and then mixing in Record. At least since 1.1 multiple audio import is not a problem anymore...
I know one might say, most of the time you already ruff mix during the actual composing and production time. But to lay out a mix from scratch can often be extremely beneficial and the end result might benefit from such a procedure.