If the ND child will not cope with moving between houses, but you also do not want to become the sole resident parent, do you think that "nesting" would be possible for you and your ex?
"Nesting" would entail that each of you and your DH gets a smaller place close to the house, and you keep the house, and your children live there full time, and on any given day, either you or your DH is staying there (but not both).
So in your situation you might have to live there for the 2/5 he is travelling for work, and in your own place part or all of the 3/5 when he is not travelling.
If one of you is bad at tidying up in the house, perhaps a jointly funded cleaner comes on the transition date, so if for instance stereotypically he is more of a slob, then the cleaner comes on the day when he switches out and you switch in, so you still get a nice clean house.
It would be shite for your schedule to depend on his - but perhaps less shite than being a solo parent? But obviously this only works if he can cope with the children without you.