Excellent thread so far!
Here's a short list off-the-cuff of things I personally consider to be non-negotiable. To be an Epicurean, in other words, is to embrace at least the following;
1. A thoroughgoing materialism. A universe that is strictly material, and strictly natural, with no supernatural element whatsoever.
2. A decisive rejection of an afterlife, and of the worry that there is anything to fear in death.
3. A full-throated endorsement of pleasure as the end or telos of human life. Number 3 seems to be the major sticking point for most.
These are not complete and sufficient Epicurean positions, but they are the major necessary ones.
If I added a 4th point, it would be the dismissal of Absolute Justice, Absolute Morality, Virtue, Duty, etc. These are, where they exist at all, conditional and not absolute.
People for whom point number 1 comes naturally find it increasingly difficult to accept 2, 3, and 4 as they move down the list. So they amend the texts, thereby counterfeiting the whole system, and pass it off as authentic.
And of course, I haven't even mentioned the Canon of epistemology!