cold feet - can recirc hot water help?

Andrew2

My toilet room has a tile floor that is cold on bare feet in the middle of the night. I am installing a recirculation hot water system and wonder if running the line under the floor will kill the chill in the spot for my feetor should I get a rug?
A floor joist bay runs crosswise to the john, right under the foot area. The tile is on an osb subfloor. If I run the 3/4" copper recirculation pipe an inch below the subfloor, place a tin sheet below the pipe to reflect the heat upward then insulate the joist bay below the tin it would give me 3 feet of pipe to heat the 16" x 36" joist bay. So would this heat my feet? The tile area is 3' x 7' so would heating a 16" x3' strip across the middle work. The water temp will probably be around 110 degrees and use a timer and thermostat to regulate pump operation.