The origin of all of this seems to be a video Trump posted on Truth Social in early May that referenced comments Kennedy made during an interview in late April. The "fake anti-vaxxer" label is a paraphrase of Trump's claim that Kennedy isn't a true anti-vaxxer due to Kennedy saying that he is against vaccine mandates but not vaccines.
In this video posted on May 9th, Trump claims Kennedy is actually a liberal and is only pretending to be an anti-vaxxer in order to split the Republican vote. Around 2:45 of that video Trump says:
... he is a radical left person, so Republicans get it out of your mind that you're going to vote for this guy because he's conservative, he's not. And by the way, he said the other night that vaccines are fine, he said it on a show- a television show, that vaccines are fine, he's all for them, and that's what he said. And for those of you that want to vote because you think he's an anti-vaxxer, he's not really an anti-vaxxer, that's only his political moment.
Judging by the time of the post, the "he said the other night that vaccines are fine" is likely a reference to this part of an interview Kennedy did on Real Time with Bill Maher on April 26th. Around 2:15 in that video Kennedy said:
-RFK: I believe if people want the vaccine that they should be able to get it. I am not anti-vaccine. What I say is that...
-Maher: Well, people think you are.
-RFK: Well, I know. But that's because I'm called at...because it's a way of silencing me. But I have said for 17 years, I'm not anti-vaccine, I just want good science. I want-- People should be able to make informed choices. I am against vaccine mandates. I'm not against vaccines.