Many white wools do not felt, it has to do with the bleaching process. The way I understood it when explained to me, is that the bleaching process removes many of the teeny little fibers in the wool, and those teeny fibers are what get all tangled up together and produce the felted material. So the bleached fibers are too smooth to felt properly.
Many cream colored or off white wools felt just fine, but I always do a test swatch.
If your bag is completely unfelted in the white area, you could use a needle and weave in another white wool (or cream) that WILL felt. Call it a designer detail. Then, felt it again! It might not be the best idea for this particular pattern, but it's a good trick to know, anyway.
I used this technique once on a striped bag, the white I was using felted poorly just like yours. So I weaved in a double strand of yellow, and it felted up nice, and due to the added yarn, it had a cool texture to it, like an applied icord "bump".
(just an idea if you get desperate!)