I decided I couldn't wait for my stuff to arrive, so I went and brought a piece of art board to use as backing and tested the stretching of what I had already stitched.
I only pinned (well, I used needles as that is all I had) the fabric around the back, but it seemed fairly easy, although fiddly, to get the design to appear square on the front. So I think I will continue on this fabric and if it can't be stretched completely straight then it'll just have to be framed without a mat - I think if there is a wider expanse of fabric any left-over wonkiness won't be noticeable. But I think from my experiments that with patience it should be possible to frame it straight.
Still think I won't use the rest of the fabric though. Although I guess Christmas ornaments and things would be alright on it.