I'm having some trouble finding what フッたん means, I can see that 降った is read the same, but I've found is never even rendered in Hiragana let alone this way. I'm not even particularly sure if フッたん is something by itself.
The full piece of dialogue is the following:
サイッテ~ ...彼氏と別れた~。
...私がフッたんだけどさ~...。~
A bit more context is that it is a girl speaking to two others.
フッたんだ is a colloquial way of saying ふったのだ.
So 私がフッたんだけどさ means "It's me who dumped him, though."
The whole dialogue is very colloquial, and it's quite common when rendering colloquial speech in writing, especially manga, to use katakana to render emphasis.
You did not make any remark about it, but the first word of the dialogue is a very typical example of this, so is a big hint when we see the フッたん later that it's still the same thing happening :
サイッテー which would normally be 最低{さいてい} ("the worst", "what a shit") shows three colloquial effects: use of katakana instead of kanji/hiragana, えい at the end changing into an lengthened え, and an extra っ before the て to mark a chopped style of speaking for extra emphasis.
