So my first question would be why do you want the double layer of batting? Is it for trapunto? You don't mention that you're cutting or trimming any of it away.
If, for whatever reason, you want to quilt the 6" square, then add borders to it, and sandwich that, you could. The double batting wouldn't really be a problem, but then the hand quilting would NOT be visible on the back of the block, right? And wouldn't you have 2 backings under the 6" part - or will you do the hand quilting without any backing?
Why wouldn't you be able to finish the whole 10" block which includes your 6" center block, then layer it, including the another layer for 6" center, and then quilt it - by hand and/or machine. And then join it to other blocks using a QAYG method?
So, yes, you can do it, but maybe there is a better way to get the same effect.