C.B.
The only comment I have is that the parents should be the ones to switch households, not the child.
They should each rent an apartment or room for themselves to stay at when not at the "family" home. Then the child stays home and the parents move in and out.
I have talked to many children who have to change households weekly and they ALL hate it! They say they never feel grounded because they are constantly moving.
My suggestion would be that they each stay one week with the child. MOther gets to be with child on Mother's Day; father on Father's Day. If they are civil, child's birthday spent with both. If they find that they cannot get along, then mother gets child for his/her birthday on even numbered years; father on odd numbered years.
Same for xmas and major holidays, mother gets for either odd or even years and father the other years. In the year that mom has the child for xmas, then dad has the child for Thanksgiving. Same for summer holidays - if mom has the child for memorial day, then dad has the child for Labor day. They switch off having the child for 4th of july - mom in even numbered years, dad in odd numbered years.
If they sit down and really think about it, detail by detail, they should be able to work something out. I do suggest not using any ambiguous language such as "at reasonable times" because everyone's idea of what's "reasonable" is different.