First, they have not claimed "honest mistake". They admitted falling below journalistic standards, publicly apologized and forced 2 of their most senior officers to fall on their swords.
Second, if GB News had done this to Burnham we have in place Ofcom to oversee regulatory compliance, so it is not a great example. I suspect Ofcom would have fined the BBC and compelled them to issue an apology. That is, pretty much, what happened here. If only the US had an equivalent organisation...
Oh wait, they do. What did the FCC do in this case? If nothing, that may tell you something about how this c0ck up was treated in the USA (or perhaps how toothless the FCC is?).
Perhaps look at, say, Fox News broadcasts during Biden's administration. Aside from the fact that (I will guess) this sort of partiality and journalistic slight of hand was a daily occurrence, I would imagine they would cite both honest mistake and freedom of speech arguments. Also, the issue would fizzle away, probably because yesterday's breach was already overtaken by today's.
Third, and accordingly, your implicit suggestion of two tier treatment of breach is misplaced.