you know, I have never done this, but I cannot think of something as easy to do as this ESPECIALLY since the home is under construction and hopefully the countertops are not installed already.
I would find a good piece of hardwood. I would then cut it to the size that you want or the old one that you like.
I would dress it down, sand it, and not add anyother chemicals what so ever to it. You can oil it with plain old olive oil only. Remember, this board will have your food on it.
Now that you have this board all squared off and prepared. I would find the logical space that you want it inserted. I would then scribe the width and thickness or trace it onto the logical location.
Make sure that there is no other obstacles under the counter that would interfer with the sliding forward and back of the board.
Cut out the area and sand and dress this slot down for easy sliding in and out. Test it for ease of operation.
Now, the next step is to install a "dog" if you will on the far end to prevent the board from comming out when you pull on it to its max reach. This can be done with a small square piece of molding. I would both glue it and screw it into place. This would be done with the board in the counter area, accessing it either from the top or from under a counter doors or something. While your under there you need to install a stop so that the board will not go any further into the cabinet and is smooth and level with the cabinet face.
The material needed is a good hardwood board, a pull, a piece for the matching cabinet, a dog, a stop, and possibly some touch up stain to match the existing cabinet.
The labor on this project should be between $75 and $100. $600 is redicous. He just does not want to do it and turning the job down with high labor costs.
Remember, do not seal or stain or paint this board, only olive oil for oiling.
Now, I would get a piece of old cabinetry from the new cabinets and cut it also the size of the board or possibly use the piece you cut from the cabinet. What we want to achieve here is a face for the board so when depressed, it matches the existing cabinetry. You can glue or screw this piece to the end of the board. I would also instal a pull or something to grab the board by.