I would say:

It is not opposite day.

Wait a second, it's already stated in the puzzle that ...

Well, let me interrupt you right there. I think the second example is a trick. The puzzle clearly states that saying something will be reversed on "opposite day", so saying "It is not opposite day" on opposite day will become "It is opposite day" which solves the problem.

Now, let me elaborate on that trick I mentionned. I think what OP meant by "If you tell me it is not opposite day, I will simply believe and agree that it is not opposite day" is that if you tell him that after the inverting has occured, meaning you originally said "it is opposite day". Also, notice that the second example doesn't look like the first example.

After all, it's a lateral-thinking puzzle.

I would say:

Today is not opposite day.

Wait a second, it's already stated in the puzzle that ...

Well, let me interrupt you right there. I think the second example is a trick. The puzzle clearly states that saying something will be reversed on "opposite day", so saying "Today is not opposite day" on opposite day will become "Today is opposite day" which solves the problem.

Now, let me elaborate on that trick I mentionned. I think what OP meant by "If you tell me it is not opposite day, I will simply believe and agree that it is not opposite day" is that if you tell him that after the inverting has occured, meaning you originally said "it is opposite day". Also, notice that the second example doesn't look like the first example.

After all, it's a lateral-thinking puzzle.