Reading between the lines, it sounds like your wall is plasterboard (aka drywall) glued directly to a masonry wall. What is the underlying wall made of?

Why take out the tee on the cold water side? Just remove the dishwasher from the tee and connect the line to the fridge there in its place. Now all you need to do is add a tee on the hot water side and reconnect the dishwasher there.

Dishwashers use only hot water, never cold. The dishwasher has it's own heating elements it can use to heat water to the correct temperature, but the dishwasher's heating system is less efficient than your water heater.