I would have to agree with jonr. I think it would work, but movement is your enemy. I saw a poly layer in your stackup, I think I would eliminate it and fasten the foam to the concrete, thinset seems appropriate. I would also do the same at the foam/cement board junction. If you need the vapor barrier I'd tape the foam before you fasten the cement board down, although there is probably one already under the concrete. I'd stagger the seams, no foam board seams on the concrete expansion joints, and offset the cement board seams from both the foam board seams and your tube channels, that way you don't create any hinge points. Last I'd skip any screws, just seems like a bad idea with the tubes under there, and I'd use a modified thinset, that stuff will stick to most things.

Fastening the foam to the concrete doesn't buy you anything- it can even be considered a slip-surface. The dimensional changes of the wood with humidity and the tile/fiber-cement with temperature will be the largest movement factor.

You could consider low heat hydronic baseboard if you have other in-floor zones in the house. I have 180 degree hydronic baseboards in my basement with no floor insulation and they seem comfortable. One room has glue down wood parquet floor tiles and the others have carpet with no pad. Cool down there in the summer so obviously the space is not well insulated from the ground temp but the additional heating costs are an ok trade off for having a cool refuge and sleeping area during 90-100 degree days in summer.