According to baidu: https://baike.baidu.com/item/喜当爹/8785247 It is use to mock/tease men who had worked really hard to pursue their goddess, but the goddess was pregnant with someone else's child and the man could only helplessly fill in/takeover/accept. Literally, it would translate to something along the lines of "Congrats on becoming a father", according to baidu, it would translate to "Green daddy" cuz there's a huge patch of green above their head o.o
Not sure if this is your worry, but as a reader, I thoroughly enjoy reading footnotes about funny slang. I'm sure I'm not alone.
You can try using stepfather as a replacement? E.g. in sarcastic lines like "congrats on being a stepfather". What's the context? Is someone saying this to another or is it a noun that's going to be thrown around a lot?
If you can find a phrase for the antonym of a cukoo bird Cuz cukoo bird "lays egg" in someone else's "nest" So someone "impregnated" someone else's "wife"
Without knowing the whole sentence, it is hard but since the MC is ridiculing himself... "I got congratulated on being father"?
I'd go with something like "I'm quite the fortuitous father" or however the sentence should go. Fortuitous generally means lucky chance but works as satire as he's deriding himself. But that's me.