The cost difference between these two is smaller than 300K in total.
“I have been accepted into the University of Washington Honors Program with a full tuition scholarship. I have also been accepted into my dream school, UChicago, but with no merit money”
So OP only gets tuition free and she/he should compare the Chicago’s tuition (52K/year) and UW’s (0).
Four years’ attendance: 220K (considering inflation).
If OP wants to graduate in less than four years, she/he may finish in three years since she/he has many AP credits (to place out many Chicago’s science Core and foreign language Core, and use as electives).
Three years’ attendance: 160K (considering inflation).
I cannot see OP graduates in two years since Bio major requires 16 credits and CORE stills requires at least 9 credits. I think you can use AP credits for up to six electives.
http://collegecatalog.uchicago.edu/thecollege/biologicalsciences/
http://collegecatalog.uchicago.edu/thecollege/examinationcreditandtransfercredit/
Still a lot of money to be involved.