I wouldn't call what I feel happiness, but (rather) an ongoing acceptance of unhappiness.
Should I include the word rather? (Or replace but with it?)
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The meaning in your example for rather is sense 5:
5) on the contrary; quite conversely ⇒ "not a help, rather a hindrance"
The meaning is similar to but, however using but tends to exclude the previous phrase.
I told you not to do that, but you did it anyway.
When using rather, this exclusion is not so distinct.
You are not saying the first phrase is wrong, but that there might be some middle ground between them.