Add Accessible Waterline shutoff for built-in style refrigerator
I did a similar install and placed the shutoff in the cabinet next to the fridge. Your idea is pretty much the same, if only a little harder to access.  If there is a cabinet next to the fridge consider that.

In any case, what I used was two, 10' braided steel "ice maker lines" (1/4" compression on both ends) with an inline 1/4" ball valve between them.  A few cable clamps were used to hold the valve in place against the back of the cabinet which makes it easier to operate, but that's optional.  One of the lines needs to be long enough to pull out the fridge and the other just needs to be long enough to reach from the existing valve to the new location.  In other words, you might not need two 10 footers - choose your own sizes.

Ice Maker Line

1/4" compression valve

(Links were the first results I found with an amazon search and only show what I'm talking about.  I didn't use those items but found identical items at a local home store.)