There were a few schools out there (and it’s been a few years so I don’t remember exactly which ones) that gave a scholarship ($1000 or so) for completing the FAFSA.
The FAFSA belongs to the student, not the parent, so it is really your kids who fill it out. Currently the parent the student lives with is to fill out the parent section but that will be changing to the parent who supplies the most support. My brother and his ex also have twins and also each take one for tax purposes, but in reality the twins are always together and the parents do not give more to ‘their’ child than the other. Food, housing, skates, skis, medical insurance is all exactly the same.