Stay Out of It
In shiurim in which someone asks the rabbi or rebbetzin about taking sides, either in a machloket between rabbanim or just differing streams of TORAH-TRUE Judaism (whether between the misnagdim & chassidim of yore, or the different chassidic groups, or any divergent derechs today), the answer was always:
"STAY OUT OF IT."
Rav Avigdor Miller constantly warned his listeners not to get involved in any way because he insisted it's getting involved in fire. Don't do it!
Rav Miller himself was a Slabodka Litvak. Yet not only did he never put down groups following other daas Torah, he even defended their right to follow their own Torah-based daas Torah, whether it was another type of Litvish (like Telshe) or Sephardi or any kind of Chassidus.
You can follow your Rav/Rebbe without fighting with anyone else and without denigrating anyone else, whether the Rav/Rebbe himself or someone from his group.
(Clarification: "Someone from his group" means someone who actually follows the rav, and doesn't just use a distorted version of the Rav/Rebbe's hashkafah to indulge his or her own yetzer hara.)
If it's a real talmid chacham, you should be able to see the value of his derech, understand WHY he is doing what he's doing...even if you don't personally follow or agree with that derech.
After exploring the issues, you should be able to come to some kind of intellectual understanding and even appreciation of a wholly Torah-allegiant yet different derech.
Furthermore, any kind of real daas Torah does not generally WANT you fighting with others, even to defend his honor.
So why do it? If you really respect him, why go against his will?
What If He isn't the Rav You Think He is?
This is particularly likely to happen among people who haven't yet gained enough knowledge to know that while their rabbi is a cool guy and a smart one, he's definitely not a talmid chacham.
I learned this lesson the hard way. (Story for another time.)
Anyway, this is something I wish I could shout from the rooftops:
Smicha does NOT mean a rabbi is wise, insightful, or good.
It COULD mean that, but it doesn't have to.
All smicha (rabbinical ordination) really means is that a guy passed a test of academic knowledge. If it was high-level smicha, he's probably intellectually intelligent.
But that doesn't mean he's "a big talmid chacham!" or anyone you should be swooning or tussling over.
(And yes, I've heard people swooning over or boasting of or defending the honor of their favorite rabbi, calling him a "a big talmid chacham" when he definitely is NOT. A smart guy? Yes. A nice, sincere guy? Maybe. Sometimes yes, sometimes no. A charismatic guy? Yes. An actual talmid chacham? NO.)
So that's another reason not to get too wound up. You might be getting wound up for no good reason at all.
Case in point: When I was becoming frum, I really thought so many people were so smart and so frum, and that certain rabbis I met were powerhouses of Torah knowledge.
Then over the years, I realized that, while all these people should be appreciated and admired and respected for their many good points, they weren't Rebbetzin or Rav Kanievsky.
And that's fine!
So while it's important to humble oneself to listen and learn from people who know more than you, there are still higher levels and lower levels of Torah knowledge & behavior, and we shouldn't make people into something they are not.
It's a tribute to frumkeit that a guy can spend his childhood and young adulthood in yeshivah, and come out with such impressive knowledge. And then as he keeps reviewing throughout life and maybe even gets smicha, he really does accumulate an impressive amount of Torah learning.
But it doesn't mean he has internalized it.
It doesn't mean he is a talmid chacham.
Getting Down to the Nitty-Gritty
And love and appreciate your favorite rav, but know what he is not (unless, of course, he really is a genuinely tremendous talmid chacham. Some are.).
And this all goes for me too, of course.