Look at Warmboard-R®.  It's a 13/16" panel with the channel for pex in it and aluminum bonded to it for heat distribution.  Then you install flooring over it. So you'll be losing 2-3" of height.  With a 7'6" room height this may not be feasible if you have tall people. 

I've not used it personally but I've heard of it being used.  One caveat, it appears to be a composite board.  I don't know how it would handle a wet basement.

See https://www.warmboard.com/products/panels/warmboard-r/