Is there another way to say "needn't have done"?

Actually, all of the examples are a little weird. Need is very rarely used as a modal any more, and it's likely that you'll be able to go your entire life without ever encountering it in speech. It is sufficient that you be able to recognize and understand sentences like the above, but don't expect to use them much. Just treat need like any other verb:

Present: don't need to

Present perfect: haven't needed to

Simple past: didn't need to

Past perfect: hadn't needed to

That said, the three examples are grammatically correct, strictly speaking. The phrase needn't have done is a fine, though archaic, way of saying didn't need to have done...

When you write I don't need to, or I didn't need to, you are not using need as modal verb; you are using it as a normal verb.

Modal verbs are different from the normal verbs because:

1. The verb doesn't take the -s for the third person.
2. To negate a modal verb, you add not after the modal verb, even in the present and past tense.
3. Many modal verbs cannot be used in the past tenses or the future tenses.

She can speak English correctly.
She should not be here, now.
She will can go with us.
She musted study very hard.

Using need as modal verb (need is used as modal verb only in negatives, or questions), you should write:

I need not [verb].
I need not have [past participle of the verb].

When need is used as modal verb, it is used to express necessity or obligation.

Need I say more?
I need not have worried.

When need is not used as modal verb (which happens most of the times), then it is used as in the following sentences:

I need to go, now.
I didn't need to know all the details.
I have not needed that for ages.

The opposite of needn't have done is should have done

For example:

It's too late to call the bank. I should have done that earlier.

Which is opposite in sentiment to

I bought oranges yesterday! I needn't have done that today.

Which can be equally expressed as

I bought oranges yesterday! I didn't need to do that today.

There is nothing wrong with "needn't have done". It means, simply, "did not need to do [something]." This is true even though the cases are different. The actions that have been accomplished were done in the past in both cases.

You paid my bill? You needn't have done that. I was fully prepared to pay it myself.

You paid my bill? You did not need to do that. I was fully prepared to pay it myself.