Why is 1 the correct choice here? I would have guessed 2 because it sounded correct to me after reading the question. I'll try to explain my thinking about all of the options here.
To me 1 sounded like it could be correct, but it could use something to connect it better to the latter half of the sentence. But it looks like just the で is enough and I need to read more.
2 felt correct because it connects to the latter half of the sentence (even if 後悔 -> してしまったことは取り返しがつかない). After seeing the answer I tried to think why 2 would be wrong. Is it wrong because XといえどもY forms a conditional ("even if X, Y is true"), and the 後悔してみた is already in past tense so it has already happened, and you can't use this kind of a conditional for something that has already been done?
Option 3 also sounded like it could be correct, but I haven't seen many cases of it being used, so I wasn't so sure. I've mostly seen it in sentences that start with なんにせよ, and not so much in the middle of a sentence. I've also seen せよ used in commands but that's also a different thing.
Option 4 is clearly wrong but I'm not sure I can explain it well. I think X ばかりに Y usually means "just because of X, Y happened", or "just after X, Y happened". It wouldn't work here because the latter half of the sentence is not a consequence of the first half.