First, let us take a look at the main clause of the sentence in question. It is 「積極的に話をすすめてもいいよ」, which means "you can/may actively pursue the matter".
Please make a mental note here that the main clause is affirmative both in meaning and grammar/structure.
Now, let us have a look at how you translated the main clause, which is "We can't assertively continue this conversation". You used "can't", but it is okay because you also used "without" in translating the sub-clause -- "without checking with the head of the department". The two negative elements cancel each other perfectly.
Thus, your translation is actually good in and of itself. I, however, suspect your own translation using two negative elements might have confused yourself in the end because in the Japanese sentence, we know that at least the main clause is affirmative.
What we would need to think about here is: What would match in Japanese the "negative + negative" structure in English when the main clause in the Japanese is already affirmative?
The answer would be "affirmative + affirmative" in Japanese, would it not? A "can't ~~ without ~~" structure would surely match a "can ~~ with ~~" structure, yes?
This is why the choices #3 and #4 are "automatically" out because with 「ない」, they are negative both in structure/grammar and meaning. Both 「ないと」 and 「ないなら」 must be followed by a negative phrase (grammar-wise and/or meaning wise), but our main clause 「積極的に話をすすめてもいいよ」 is 100% affirmative.
So, we are now down to two choices -- #1 受けてからだと and #2 受けてからなら.
1「受けてからだと」 should normally be followed by a phrase that is negative either in grammar or meaning, which is why 「受けてからだと話をすすめてもいいよ」 sounds very awkward. It is not something careful speakers would say.
2「受けてからなら」 fits perfectly into the sentence. "If it is after getting the manager's OK, ~~" It finally gives us the "positive + positive" sentence structure that we were looking for.
I admit that careless speakers (native speakers, I mean) might have chosen #1, but the sentence using #1 would be corrected by every teacher or boss.
Choices #3 and #4, unfortunately, are just utterly impossible. They both need to be followed by negative phrases.
With #3, you would be saying "Unless it is after getting the manager's OK, you may actively pursue the matter." Makes practically no sense, right?
With #4, you will have "If it is not after getting the manager's OK, you may actively pursue the matter." Equally nonsensical, yes?