I’m not sure if it’s the easiest way, and I’m not sure it’ll work for this particular instance, but…
I would duplicate that outer shape and the intersecting bezier. Then temporarily change the color of the new shapes and hide that layer. Then select the remaining outside piece, convert it to a path if it isn’t already, and enter node edit mode. Once there, I would delete, for instance, the right half as close to the end node points of the intersecting line. Select both, go to Edit -> Auto Join Selected Shapes. That should create one solid shape.
Repeat those steps for the other side, and the end result should be you having 2 solid shapes that you can then fill as you like separately.
I hope that’s easy to understand, as I haven’t even had a half cup of coffee yet. If not, I can add some pictures to try and illustrate my point.