6 courses in your senior year doesn't seem like a great solution. If you figure you take Math, Science, History, English in your senior year, Then take 2 electives, that is basically a full schedule already. Short of replacing one of the poor grades with a freebie like Gym, I am not seeing where that solves the issue. This might be a difference in US and Canadian education, so maybe 6 courses means something different.
A mathematics-focused student in Ontario could take Calculus & Vectors, Advanced Functions [1], Computer Science, and Physics—the first two courses should be straightforward for this student.
Though Computer Science and Physics are distinctly different from the mathematics courses, these are still directly useful for a mathematics student to learn—the problem-solving skills should also carry over. Key mathematical discoveries have been inspired by problems in computer science and physics, and many rigorous university-level mathematics books still draw from problems in these fields to motivate certain problems. At the least, they are less laboratory-heavy than Biology and Chemistry (the student could still attempt these subjects, though, and choose to omit the grades for university admissions).
That leaves a couple of other classes—or just one if English is required, as noted by another commenter. My school offered subjects like Grade 12 Drama, Visual Arts, and Music, where much of the grading was effort-based. In my school, most students in my classes saw these courses as a break from other intensive courses, with grades not being as much of a concern. This would allow the student to avoid using a grade for History, Economics, French (or another foreign language), or another subject.
The English requirement would then be a difficult challenge for the mathematics-focused student. I wish I could speak more about what it was like for most of my classmates who went on to study engineering, as many of them took the standard English course (I took a more demanding version of the course, due to personal interest). My classmates at the time did not seem to have an issue with university admissions to competitive programs despite not enjoying the subject at the time, but the other commenter makes a good point that minimum grades for admission standards have increased greatly since then.
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[1] As an aside: a past classmate—who was brilliant at mathematics and also great with people—later poked fun some years later about the Ontario government's naming for math courses. He said, "there's Grade 11 Functions... and then in Grade 12, there's Advanced (!) Functions." The last I heard, he went on to work as an investment banker at a top hedge fund by profitability in the United States.
The first math course as math major in university involved proving that a+b=b+a and that a+(b+c)=(a+b)+c. It's quite fun to go from 'advanced' calculus during the final classes of high school, to "OK, let's consider the expression 1+1=2. What does it actually mean and why is it true?"
You can get tested into qualification, by most colleges. Colleges (which includes Universities, State schools, etc) aren't robotic. This is part of the responsibility of the administration. You may get delayed a semester or a year, with a little community college gatekeeping. Dedicated students can always get into a University.
