I was hoping someone with more experience would speak up.
I've never had to repair damage where a part of the cone is missing. It looks repairable to me. If that were my speaker, this is what I would do:
It looks like the cone is a paper material. I would find some material of the same thickness, rigidity, and weight as the cone material. If you have another driver with similar cone material that is unusable, I would use the cone for a patch. You do not need much, so even buying a cheap clock-radio from a thrift and cannibalizing the speaker would work.
Make a patch the same size as the missing part of the cone. I would ensure the mating edge is rough, just like the edge of the missing cone section. I would then make a backing patch from a section of coffee filter slightly larger than the patch. The cone patch fits into the missing section and the filter paper patch goes on the back side of the cone to bridge between the patch and cone.
I would use Aileen's Tacky Glue to glue the edge of the patch to the cone and to glue the filter paper to the back of the cone. I would use wax paper on each side of the cone to squeeze the joint for a few minutes so the glue will set. This will ensure the cone patch is even with the rest of the cone. With the glue set, install a new foam. The foam will help hold the patch in place and provide most of the support for the patch.
I do not think that the weight of the repair would unduly affect the driver. The cone patch should not be adding weight because you are filling a hole. The only added weight will be the filter paper patch and a small amount of glue. That is not really much weight. The shape of the missing section is shallow relative to the edge of the cone, which is good.