It is unclear if he was symptomatic in the first positive test. I have read at least some of the tests are capable of giving false positives. Could this have been the case here?
There are no free wins here. My advice follows some of the others: during work hours try to schedule time and have articles/books/videos queued up for downtime after kids are asleep when your SO isnt needing your presence/company and you are truly up for it.
Trying to get up early and hack your life will just lead to burn out and family problems both. Dont do it to yourself, your SO or your kids.
Be present with your family out of work. Squeeze in 30min each morning when you get to your desk to hack on something you read on the train in or at home. Accept your situation and also realize that 30 min daily spent purposefully is enough to make real progress in one direction at a time.
It is exhausting with two small kids and an SO and a job. Please dont kill yourself trying to be super human. Accept limitations and live a better life within them!