Quoting the Sefer Haikkarim is a big mistake. He doesn’t argue on any Ikkar of the Rambam, Chas Veshalom. He argues on the number, because he held that they are either included in the first three or should not be considered an Ikkar, even if one must believe it.
You can’t disagree that trying to bend Judaism in a new direction does not follow the attitude of our earlier Rebbes. You might not care about that, but that should draw the line if it is a new movement or not. If it is a new movement, it should get a new name.
I cannot argue about individual Shitos because I never cared enough to read up on them.
As to poking fun, perhaps it is indeed wrong, unless you find them aggressively stepping on Yessodos. But it has nothing at all to do with being Yeshivish or not. Yeshivish is not the point.