Plus even that isn't enough: 10/3 has an infinite decimal expansion (in base 10 at least) too, but if π = 10/3, you'd be able to find exact circumferences. Its irrationality is what makes it relevant to this joke.

A mathematician is also perfectly happy with answers like "4π" as exact.

Plus what's to stop you from having a rational circumference but irrational radius?

Writing this, I feel like I might have accidentally proved your point.

Its decimal expansion is finite in the base pi.

This is the correct answer. Pi is known. What it's decimal expansion looks like is irrelevant. It's 1 in base Pi.

The actual punchline here should have been “there is no known equation to calculate the exact perimeter of an ellipse”, then sucking tears from an astrophysicist

Exactly, a fraction is completely as valid of a way to express a number as using a decimal.

1/2 = 0.5

They're both fully valid ways to write the exact same quantity

This was my first thought and then I realized I had been nerd sniped.