Here is how I would define axiological hedonism:
- Suffering, i.e. any negatively valenced experience, has intrinsic disvalue.
- Pleasure, i.e. any positively valenced experience, has intrinsic value.
- Nothing else has intrinsic value or disvalue.
The core of my take on axiology is that something has intrinsic (dis)value if and only if it literally is valuable. Intuitions are not evidence for intrinsic value.
I am convinced that suffering (and pleasure) fit this criterion. The disvalue of suffering is self-evident from introspection, i.e. from observing how suffering feels. The disvalue is inherent in the experience; it is not a matter of an evaluation done by me, or a desire for the suffering to stop felt by me (even though there is a strong correlation), or me having a certain attitude towards suffering.
That being said, I think a subjective judgment cannot be avoided when it comes to comparison of (different kinds of) suffering and pleasure. In the words of John Stuart Mill:
Neither pains nor pleasures are homogeneous, and pain is always heterogeneous with pleasure. What is there to decide whether a particular pleasure is worth purchasing at the cost of a particular pain, except the feelings and judgment of the experienced? (What Utilitarianism Is)
An experience does not inherently carry information about its commensurability with other experiences, but it does carry the basic information that its value is positive or negative. The basic variant of axiological hedonism is only concerned with the latter.
See the Theories of Well-being chapter at utilitarianism.net for a detailed philosophical overview of this topic.
The simple case against hedonism is just that it is bizarrely restrictive: many of us have non-hedonistic ultimate desires about our own lives that seem perfectly reasonable, so the burden is on the hedonist to establish that they know better than we do what is good for us, and - in particular - that our subjective feelings are the only things that could reasonably be taken to matter for our own sakes. That's an extremely (and I would say implausibly) restrictive claim.
When you say "what is good for us", could it be translated as "what we are attached to"? If you care about knowledge or relationships, you will experience (dis)satisfaction depending on what relevant events happen in your life, and you will be motivated to achieve goals related to these things, but this is a far cry from what intrinsic value means in my view.
In essence, when I say suffering is intrinsically bad, I don't mean that it is bad for anyone; I mean that it is bad period. The badness is an inherent feature of the experience.
So from my perspective, the non-hedonist is making an extraordinary and unfalsifiable claim when positing the existence of non-experiential goods.