One of top engineering institution in my country offer Bachelor of Sciences in Mathematics. They start actual mathematics (Real analysis, Abstract Algebra etc) from sophomore year. By the senior they offer graduate level classes and I am sure I won't run out of it. However this school is primarily known for CS undergraduate program. Every year one or two graduating CS majors go to the best graduate programs (CMU, MIT, Stanford etc). As I want to major in pure mathematics I am pretty skeptical how top graduate programs (Princeton, MIT, Harvard) will view my application as they will be unsure of the rigor of Math degree at my institution?
If you're doing proof-based real analysis and algebra by your sophomore year, you're doing superbly. Many well-regarded schools don't introduce such coursework until the third year, opting to spend four full semesters on calculus. Sometimes the ultimate or penultimate semester is replaced by an intro to proofs, discrete mathematics, or linear algebra course, but the effect is the same. One of the major draws for me going to my undergraduate institution was that the norm was for mathematics majors to do real analysis their second year, with exceptional students starting little Rudin on day 1.
The extra year of advanced mathematics courses will be a huge boost to your application. You can increase this effect by demonstrating breadth and depth that other students won't be able to match. If you spend your first three years accomplishing the theoretical coursework that most students do, and then spend your fourth year taking, e.g., the graduate sequence in algebra, graduate courses in discrete mathematics and algorithms, and advanced undergraduate courses in functional analysis and algebraic topology then you've effectively demonstrated your ability to succeed in graduate level coursework. You'll also have a far deeper understanding of algebra and it's applications to computer science than the vast majority of applicants, and taking more advanced courses in analysis and topology lets you show that your advanced status isn't shallow or restricted to one area. And just in terms of a numbers game, you'll have ~12 advanced mathematics courses to other people's six (please correct these numbers if they don't make sense; I've only attended quarter systems schools and am not 100% on what numbers are normal).
Breadth and depth of coursework is a commonly used proxy for future mathematics success, and if you're able to take these fourth year graduate courses with professors you had already, then in their letters of recommendation they'll be able to talk about how you excelled in their previous courses and that they're excited to have you doing graduate work and that they expect you to succeed. Comments like that can go a long ways.
I don't think that you should be too caught up about the reputation of your university. In my experience talking to people who have served on admissions committees (note: I have never done so personally) is that, if they explicitly think about the reputation of a university at all, it is in very broad strokes that encompass dozens if not more universities per tier.
If this is an issue that concerns you particularly, you might find better success asking people at your institution who are more familiar with where you go to school than I am. One thing you should definitely ask is about what graduate schools other students like you have attended recently. This last bit is very important as, especially at a larger school, your outcomes and those of the average student are likely to be far apart.
In addition to talking to members of the mathematics faculty, you should also consider talking to any CS Theory faculty that you have. If you suppositions are correct, they're probably as likely or more likely to have written letters of recommendation for students who went on to top mathematics PhD programs. A sizable chunk, if not most, CS Theory people have mathematics PhDs rather than CS PhDs.