These doesn't appear to be a motive for a Canadian birth, IMO. There was no “free” healthcare in Cananda until after Barry was born and I do not recall any accounts of lefties going to Canada to have babies.
There was at least one forged BC put on the net for Barry which was quickly debunked.
A cousin of SADO lived on the border, but her family objected strenuously (even seeming to protest too much) to suggestions that SADO was there and popped over the border to have a baby.
At the time I am sure that Seattle had plenty of homes for wayward mothers available, like the rest of the US, and SADO could have stayed in touch with the friends and relatives there...the ones that took her around to visit her friends in late August 1961 where they provided the eyewitness accounts of seeing Barry (with varying consistency, to be sure).
British Columbia and several other provinces had province-wide universal hospital care by 1950. The federal government jumped in starting in 1957. I don’t know what the policy was for aliens, but I do remember some people talking in the mid 60’s about having got free hospital delivery by going to Canada. (Not just for draft dodgers.)
It’s just a thought.