You can do blood tests to catch cancer mutations, used presently. Many blood tests.
It can be a cancer mutating. It can be the cancer not mutating (occurs after drug given) but is mutated (mutation occurred already). We have a population of a billion cancer cells. Treat with drug. 999 million die, 1 million survive because they had that mutation conferring "fitness" i.e. resistance to drug. three months later, we have a billion cancer cells again, descended from the 1 million cells. Now those billion cells also get the next line of therapy. 999 million die, 1 million survive. 3 months later, those 1 million are now a billion again. And so on. Point being, I'd think of it more as a selection mechanism as Darwin taught us, not the cancer automatically generating a resistance mutation - the blind watchmaker. In this case there was probably a mutation changing or downregulating the antibody target, probably HER2 or something related.
I don't know what "rejecting the antibody" means. You would need to look up the ADME to understand physiology and think about how the cancer might modulate that.
All this talk of "oh how do we catch these mutations".. ... ... there are a few dozen companies that will tell you what mutations are there straight up, just from the blood. From the tumor, any hospital can tell you all the mutations. The problem isn't that we don't know. We can tell you the mutations. We can tell you the new mutations. So we find 100 mutations let's say. Okay. Next question, for each of the 100 mutations, does it cause resistance, yes or no? Do two of them together cause resistance? Are they from the same cell, they could be from distinct tumor lineages. You know most of the mutations are what are called "passenger mutations" right? Red herrings.
So this is the pathology of computer scientists. "If only a clever programmer took a crack at this, these biologists what with their humanities style miasmas and rote memorization topical field". Many have in fact... Bioinformatics goes as far back as the 1970s I'd say if not further. And that clever programmer did find all the mutations. Did a great job. Pretty well developed. Then you say "ah let us understand what the mutation does!" Okay. So now you're taking a subset of the broader field of genomic variation and computationally deriving how that variant influences trillions of different cells interacting with an antibody protein with a chemical bound to it. Congrats, if you're such a "clever programmer" than by solving this, you've solved life itself! Basically this notion is "this looks easy, what is this, like the 3-SAT problem, figure if someone was clever enough to take a crack at it then that would solve this whole 3-SAT issue!" completely blind to the fact that it's just as hard (and the same) as proving P=NP. So if you ARE a hardcore clever programmer, then this rabbit hole goes as deep if not deeper than P=NP, and the cancer will humble you, as this is what cancer does, to humble.