Is the construction in bold correct in this sentence:
It was changed just a month after you had joined, so you couldn't have got used to that too much.
You couldn't have got used to is fine, but in US speech it would be gotten, and I think in conversation you'll more likely hear You couldn't have gotten too used to that. We try to keep modifiers as close as possible to the terms they modify.
Assuming that they're co-referent, I'd flip the that and the it, or make both pronouns it. The that points to something outside the sentence, so it should come first; it then establishes a convenient referent for it.
The perfect construction in the first clause is unnecessary as it stands—see here and other questions under the past perfect tag. Simple past will do:
It was changed just a month after you joined, so you couldn't have gotten too used to it.
But if you flipped the two time references it the perfect would be just dandy:
You had only been here a month when that changed, so you couldn't have gotten too used to it.
Is the construction in bold correct in this sentence?
Yes. There is nothing wrong with it. Why would you think that there was? Using the word got like that, is correct in England and the rest of the UK.