Learn Finnish: En halua tulla sivuutetuksi, vaan oikeudenmukaisesti kohdelluksi. - I do not want to be ignored, but rather treated fairly.

Using tulla with a passive past participle in the translative case (-ksi) specifically emphasizes the personal experience of the subject. While a standard passive clause like että minut sivuutetaan is perfectly correct, this tulla construction translates much more neatly to the English "to be [verbed]" and focuses heavily on what happens to "me" rather than just the action itself.

I know the passive past participles for these verbs are sivuutettu and kohdeltu. Why do they change to sivuutetuksi and kohdelluksi?

This is due to consonant gradation! Adding the translative ending -ksi closes the syllable right before it (e.g., -tuk-si, -luk-si). A closed syllable forces the preceding consonants into their weak grade. For sivuutettu, the tt weakens to a single t (sivuutetuksi). For kohdeltu, the lt combination weakens to ll (kohdelluksi).

Why is vaan used here instead of mutta for the word "but"?

Vaan ("but rather") is specifically used after a negative statement when you are presenting a direct, contrasting alternative. Since the sentence starts with the negative En halua (I do not want) and immediately offers the exact opposite alternative to being ignored, vaan is required.