I want to preface this by saying that I read zanman60's answer, but it's very slightly different than mine. I somehow did arrive at the conclusion again, I thought it was my own until I went back and saw his... well, here's my justification, although much of it repeats. Consider it in support of zanman60's if you don't think it is distinct.
You are a...
KV2 heavy artillery tank probably, or some other Kliment Voroshilov later model that has the large gun. So not the original KV1, as zanman60 seems to say...
My body is top-heavy, as my weapon is deadly.
The tank has a really big turret on top, disproportionate to its chassis. It's top heavy.
To make my shot count, faith is needed in large amount.
Not sure and the OP says it is obscure, but it could have had a recoil problem or something related, so that it was difficult to fire. It should be noted the KV tanks were not "ergonomic," so it was hard to see where to shoot.
A high profile target I may be, yet the white death never came for me.
The tank is high-profile, because of its large turret (in other words, it sticks out). As explained in Bass's answer, the White Death is not a reference to Tuberculosis (which is what I thought at first,) but an extremely deadly sniper. He didn't target tanks. It could also be you are referring to the fact it is named after Kliment Vorishov, a high-profile politician in the Stalin era of Russia.
When the greater fight broke out, my tardiness can't be bailed out.
The tanks were effective, but very slow. They also couldn't turn the gun, except on level ground, because of the very large howitzer gun. They were expensive to produce and useful when the Germans couldn't pierce their armor, but they got wise fast during the invasion. What do I mean by that? See the second hint's explanation below, where operators couldn't "bail out."
So people get bored with my dull thriller, replaced me with the pipe organ and the neckless killer.
So the Russians got new tanks, because they couldn't fight with the KV2 after the beginning of the invasion of Russia. zanman60 explains this some, where the "pipe organ" is not original to his post and represents the Katyusha rocket launcher (mounted on trucks, I suppose). I'm not sure about the "neckless killer," and google is utterly baffled by the phrase. I would argue that it could refer to the T34, which replaced this tank in later battles. zanman60 can't be right that this is the "elder brother" in the fourth hint, because it was made after the KV1 and KV2 tanks were shown to operate poorly in battle.
Hints: (on par with zanman60's, for the most part)
I'm not sure what the video game is, World of Tanks is as good as any @zanman60. It sounds obscure, as the OP tells it.
In the battle of Raseiniai in 1941, exactly the situation described in the molotov answer occurred where tank operators couldn't "bail out" after German's threw grenades into their tank. Specifically, there was a skirmish with an extremely deadly KV2 tank, likely taking over 24 hours, where the tank finally met its end in this way. This is also a similar situation to the ending scene of the movie Fury, although the tank in that movie was not a KV tank, and the movie is set near the end of the war (well after 1941). So actually, this is a bit misleading also.
This is accurate. This hint and the fact it's a tank were enough to get the solution from a google search. So I feel justifying this one is somehow backwards. See the blurb from wikipedia for the citation.
The brother is older, so a tank produced before the KV2. Likely the KV1, which has a totally different turret mounted on it, which is much smaller. Therefore, the difference really is the "head." So I suppose I just flipped the two other tanks in zanman60's answer... but this makes a bit more sense I should think.
As my final support, see these two blurbs from Wikipedia.
A heavy assault tank with the M-10 152 mm howitzer, the KV-2 was produced at the same time as the KV-1. Due to the size of its heavy turret and gun, the KV-2 was slower and had a much higher profile than the KV-1. ... Few were produced due to its combat ineffectiveness, mainly the decreased speed due to the weight of the new gun and turret. Due to an increase in turret weight from the expanded dimensions and a heavier gun, the turret traverse mechanism could work only on level ground. (KV2 tank description).
Second blurb, a caption of a picture.
The KV-2 heavy artillery tank's 152 mm howitzer was housed in an enormous turret. This prototype differs from the production version in several ways. It was called the Dreadnought by its crews.
I have to admit, I really came about this answer by seeing the type of answers that were supported (including the "pipe organ") and more or less googling that with the third hint. Seems like we'd have had to know a lot of history otherwise...
Because so much of my answer is the same as zanman60's, if this was the technically correct answer, he should most definitely get the bounty.