What if $\pi$ was an algebraic number? (significance of algebraic numbers)

No such universe is possible, it would be a universe in which $1$ is equal to $2$.

That said, a rational approximation to $\pi$ with error $\lt 10^{-200}$ is undoubtedly good enough for all practical purposes.

Lindemann's proof that $\pi$ is transcendental was a great achievement, but knowing the result has no consequences outside mathematics.

You have to understand that although $\pi$ is a real number, it's not actually a real number. That is, it's in $\mathbb{R}$, but that set does not exist in the physical universe. It's an abstraction, just like the imaginary number $i$ is an abstraction, and one that has found significant use in physics (quantum mechanics and electrical engineering among others). Just like the idea of a number at all is an abstraction: the abstraction of assigning the same description to different quantities that are not directly related.

My point is that the quality of the universe that allows such abstractions to be imagined by intelligent creatures seems not to be separable from the quality that allows intelligent, imaginative creatures to exist at all. It requires only a sufficiently descriptive language, such as the kind considered in mathematical logic, to write down the formal definition of $\pi$ and indeed, of all of our mathematics, which implies all the algebraic and analytic properties of $\pi$ that we have proven because we wrote the proofs in that language!

So no such alternate physical universe can exist. On the other hand, one could imagine basing the definition of $\pi$ on alternate axioms, such as those specifying a particular non-Euclidean geometry, in some of which one does have $\pi = 3$, say. At least for some circles.

Interesting that nobody has mentioned: A practical consequence is that you cannot construct $\pi$ using a compass and a straightedge. This has saved so-so many man-hours; if Lindemann hasn't proved $\pi$ were transcendental we wouldn't have, e.g. caramel macchiato (or more significantly, aircraft).