It takes time to enroll in a new school, especially one outside your state of residence, more so in 1961 than today. You have to apply for admission, get accepted, etc, etc.
Even in state, I had to start the process in the winter of '67-'68 to enroll in fall of '68. I doubt U Wa was any more efficient, than U. Nebraska. I know Oklahoma was still less efficient than Nebraska had been in '67-68 , 7+ years later. Although enrolling as special, non degree seeking, grad student was easier than undergrad or regular grad student. She didn't stay in Seattle because whoever was paying the bills and calling the shots wanted a record that he was a citizen in Hawaii and thought it was important to get him there.
Washington state, or any other state would have served just as well for making him a US Citizen, which would have been the objective of any scheme to show him born in the US, rather than someplace outside the county. In '61 even commies thought being a US citizen was a worthwhile thing. It was just easier to "fake it" in Hawaii, for several reasons, one being Grandma Toot's position in the area.
Most likely she didn't stay in Seattle because, BHO Sr, was no longer in Hawaii, she had money troubles, and probably Grandma and Grandpa Dunham were re-thinking the notion of being isolated from their grand-baby and daughter. Time heals those sort of wounds. I know it finally did for my wife's family when her sister married a man of a another race, and had two children with him. Believe me, not only is he the best of her three husbands, he's a hell of a nice guy, unlike BHO Sr, who was a womanizing drunk.
The agreement to give up the baby was more likely something Toot came up with. Stanley Ann could go to Washington, which is what she wanted anyway, in exchange for going elsewhere to have the baby and giving it up for adoption. Under this theory, the friends in the area, Toot's friends, would have helped locate a place where she could have the baby and which specialized in "unwed mothers", even if S.A. wasn't unwed, and finding homes for the babies. There is just such a place "Grace Hospital" in Vancouver. The friends may possibly even have provided a place for Stanley Ann to live for the last months of her pregnancy.
Even though classes at U Washington didn't start until late September, one would need to show up a few weeks early to find lodging, probably go through class registration, and otherwise settle in, before classes started.
Questionable what they did do but I think they took him back to Hawaii because the planned to do something to report his birth that required his presence
The official Hawaii birth certificates of that era did not show footprints, or anything else that would positively ID a person as being the one described on the certificate. Normally they had the hospital and doctor's name (and signature) but even that wasn't required if the birth was reported as being "at home". All that was required was a signature of a witness in addition to a parent's signature, (usually the mother's). You did not even actually need to prove there *was* a baby at all. You could create a false person, if one had some reason to do that. However the BC would clearly indicate a "home birth" and that alone would be somewhat suspicious given the 'station' of Stanley Ann's parents, particularly her mother. Something a "native woman" or a "farm worker" might do, but not the child of a banker. It wasn't cool in '61.
The decision and enrollment at Washington in 1961 could have been made in mid September. I have first hand knowledge.