There's debate, obviously, but that's how I'd do it.
At the start of your next round, the opponent is not denied his Dex bonus to AC. At the point where you make your normal melee attack, he
is, and so he fulfils the requirements for Expert Tactician - an opponent who is denied his Dex bonus. You are now eligible to make an extra attack against him, either before or after your regular action. Since you've already started your regular action, it has to be after.
So, when you make your ET extra attack at the end of your round, the Feint has "worn off", and he is not denied Dex.
Alternative readings:
1. The opponent must be denied Dex
when you make the extra attack. So you can't do it before - he's not denied Dex until you attack, so you don't have the option of using your ET attack. And you can't do it after - he's no longer denied Dex once the Feint is "used up". So Bluff is of no use with ET; ET is only useful if the opponent is flat-footed, or blind, or balancing, or stunned - something that leads to a
persistent loss of Dex, not a fleeting loss like Feint provides.
2. The extra attack must be one such that the opponent is denied Dex against that attack. So you
can use your ET attack at the
start of the round - or indeed at the end of the first round - because that will be the "next attack" that the Feint causes him to lose his Dex against. You
cannot use it at the end of the second round, because if you did, he would retain his Dex bonus, and would therefore be an invalid target.
I personally use the "trigger" interpretation, but there's disagreement
-Hyp.