For those in a similar situation as OP:
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not all US boarding schools are full. Go to Boarding School Review, look at schools with rolling admissions, and email them to inquire if they still have openings.
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There are numerous Canadian boarding schools worth looking at. Some have different admissions dates. All you have to do is email them to inquire. They are going to be full-pay, but generally cheaper than US schools.
Brentwood College School (BC)
St. George’s (Vancouver)
St. Michael’s (BC)
Shawnigan Lake (BC)
Stanstead College (Quebec)
Bishop’s College School (Quebec)
A bunch in Ontario (including Ridley, Appleby, Trinity, Lakefield, Ashbury, others)