What? Of course it isn't! I mean, I guess you could argue that it's technically a slur in the sense that it is disparaging, but obviously the guy knows that he's being disparaging- you use that word to draw connection to identitarian slurs which are inherently wrong, and bureaucrat is not that. It also refers specifically and exclusively to the parts of the government that aren't democratically elected; the opposite of what you're saying.
Even setting that aside, 'the fat cheeto and his deplorable clowns in congress' is a slur for a democratically elected government, "the will of the people". So what? We shouldn't be allowed to insult a democratically elected government for some reason? Democracies are certainly preferable to autocracies, but that doesn't mean 'democratically elected' is a synonym for 'good'.
"you use that word to draw connection to identitarian slurs which are inherently wrong" you are reading into this too much, slur is often used as a word for a general insult.
Doesn't make a lot of sense to me to respond to 'I don't want to deal with their bureaucrats' with 'You do realize that's an insult, right?'. Yeah, he realizes, he is clearly trying to be insulting. It's only an insult because of that intent, in fact. A lot more sensible if the intent is to suggest that the word ought not be used because it is an insult beyond what is acceptable in polite society, which is the much more common usage of 'slur'.
Not to say it's impossible you're right that it's being brought up irrelevantly, but I do think the odds are on my side and I further think it would be worth writing a sentence calling that out even if they weren't.
His point, I assume, was that many people insulting beurocrats think that those are somehow seperate from the people they elected i.e. it's not some unidentifiable blob responsible for these things but the person /you/ voted for. At least that's my charitable interpretation.