that's probably a lot higher than the sort of atomic number we can realistically reach in the lab?

Oh, it's immensely massive compared to anything that's likely to possibly exist as an element. But in proving something is finite, it doesn't matter if we overshoot by a million million percent, all we have to do is establish any bound at all. We don't have to engage with the question of what we can do in a lab, because the question only requires any theoretical limit to exist.

It may be difficult to predict just how far we can go in a lab, but the existence of black holes means there must be some limit.

nonsense . heavy nuclea are bound states of many constituents . no different from grain of salt . grain of salt can be much heavier than planck mass (which is about few micrograms

nucleus is extremely fluffy by general relativity standards . for black holes the mass and size are linearly proportional . the solar mass black hole has few mile size and has density few times smaller than that of the nucleus . solar mass is about 10^57 protons or about 10^55 heavy nuclea like uranium . so in order for general relativity to be important for uranium nucleus, it has to have a size of 5 miles/10^55 or about 10^-49 cm . instead heavy nuclea have a size of about 10^-15 cm . so nuclea are 34 orders of magnitude fluffier than they should be for any general relativity to be of any relevance . they are as good as grain of salt