Not a full answer, but possibly the start of one. I came across a paper discussing some mathematical properties of fullerene graphs. In section 4.2, they give a formula for a lower bound on the diameter based on the number of vertices $n$, assuming there are only pentagonal and hexagonal face.
Given a lower bound on the diameter, it may be possible to prove that a ball of a certain size would be mechanically unstable. I suspect this would give a fairly high upper bound on the size and it wouldn't account for the wide variety of shapes that are possible for fullerenes, but this would at least be a starting point.
This could hopefully be refined somewhat by using the largest synthesized/reported fullerene size as a lower bound. Based on , you will need to look for some kind of mechanical/kinetic stability limit, as it suggests that fullerenes should increase in thermodynamic stability as $n$ increase, approaching the structure of graphene or a infinite nanotube.
- Vesna Andova, František Kardoš, Riste Škrekovski. Mathematical aspects of fullerenes. Ars Mathematica Contemporanea, DMFA Slovenije, 2016, 11, pp.353 - 379. ffhal-01416354f
- WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207