I'm going to make quite a few assumptions, and include a significant disclaimer to my answer.
First, the disclaimer: I will not list a practical carrying weight, or even an estimate of such, as I lack the mathematical skill and biology background to attempt that. But I believe I can contribute to the question anyway. Possibly useful comparisons here.
Now the assumptions:
1. "Hand held" I understand this to mean something intentionally designed to be primarily carried and used by a single person, as opposed to just anything someone can manage to pick up (like removing a mini-gun from a helicopter or a 50 caliber from a hum-v, etc).
2. "post apocalyptic" In any scenario I can imagine that could be termed an apocalypse, any militaries that were not immediately wiped out would be very active immediately following the event, and would therefore be actively using their resources, including weapons and ammo. So 'post' apocalypse, the amounts of specialized military equipment available, including more unusual firearms variations and the accompanying ammo (heavy caliber, tracer, armor piercing, etc.), would be negligible. This leaves only the weapon and ammo types that are available to either civilians or the 'run-of-the-mill' 'grunt' soldier, so something on the scale of a big-ish hunting rifle (or military equivalent) is about the most dangerous the mammoth would face.
3. "anything an elephant can do a mammoth can do better" at least as far as the desired information for this question is concerned.
4. "could be of a different temperament and intellect than the historical species" (from comments) To me this means my answer can disregard the animals' reactions to the chaos of battle as unrelated to the question and as a topic to be addressed by the OP some other way.
Now, there is a reason that "elephant guns" were called ... "elephant guns". Elephants don't go down easy (compared to most other animals), even with no armor at all. Granted that weapons technology has improved since the era of elephant gun use, modern hunting rifles can still have trouble with knocking down something as small as a deer with a poorly placed shot. And even a (eventually) fatal shot on a deer can leave it alive long enough to run for miles before collapsing. Accounts tell of elephants taking up to 35 rounds before going down, again with no armor at all.
So, a Mammoth would (theoretically) be even more resistant and durable, even without armor. The weapons it would be facing would not be specialized to the point of significantly overriding it's natural defenses (even a few well placed, body mass, but not brain or heart, shots on an unarmored Mammoth will not bring it down, and it may even make an effectively full recovery). So additional armor, such as salvaged Kevlar vests, would certainly help, but would not need to be particularly highly defensive by themselves. Rather they would complement the Mammoth's already significant defense. What would protect a human from only the smallest and slowest of calibers, on a mammoth would make the smallest and slowest rounds a non-issue (other than things like shooting out the eyes), it would turn medium 'level' rounds from a flesh wound in to a bruise or pinprick, and it would turn injuries from heavier calibers like hunting and assault rifles from severe internal trauma in to a graze.
So the armor intended to protect the mammoth itself would not need to be of (relatively) significant weight. If carrying a structure meant to protect driver and passengers, the armor there would be the heaviest weight requirement other than the passengers themselves (I imagine mounting the remains of an armored military vehicle, like a hum-v, on it's back). If an elephant can carry the weight of just the visible bodywork of a normal car and passengers, then a similarly structured animal should be able to carry just the armor portions (no motor, drive-train, wheels/axles, or other unnecessary heavy components) of a small derelict military troop carrier originally built with plenty-enough armor to withstand most small weapons discussed here.
And there you have it, no exactly defined weight, but certainly an answer to "is it plausible", and some useful points to consider along with it.