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The warforged integrated protection uses your proficiency bonus as part of the calculation for your AC at all 3 levels if you are proficient in that armor type. The Defensive Duelist feat allows you to add your proficiency bonus to your AC as a reaction when wielding a finesse weapon and you are attacked. Now I know normally you can't add your proficiency to something more than once but I was wondering if this instance might be an exception because otherwise this feat is completely useless to warforged.

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RAW, they don't stack

From the basic rules on proficiency bonus (emphasis mine):

Your proficiency bonus can’t be added to a single die roll or other number more than once.

In this case, the number would be your AC for this attack. So, a warforged benefiting from Integrated Protection and using one of the modes that already add your proficiency bonus to AC would receive no benefit from the Defensive Duelist feat.

As suggested by Darth Pseudonym, note that there are times when the game intends you to apply the proficiency bonus more than once, such as with the Rogue's Expertise, but the language they use in those cases is "Your proficiency bonus is doubled", rather than "add it twice", so you are still adding it once, you just doubled it first.

Even in such circumstances, note what is reiterated further ahead on the proficiency bonus section:

If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.

Unless a feature specifically overrides this general rule, proficiency can only be added once. Neither the feat or the warforged's Integrated Protection override this rule and therefore they do not stack.

It's not completely useless to a warforged

The Integrated Protection feature for warforged is simply one of the ways they have available to calculate their AC. If they gain other ways to calculate AC that do not add their proficiency bonus (being a Draconic Sorcerer, for example) they can use those instead and then benefit from the feat.

Being a bad feat is nothing new

A feat being bad for a character is not really a reason to make an exception. There's plenty of feats that are just strictly worse than an ability score increase for characters, notably things such as Moderately Armored on a character that is already proficient in medium armor and shields. With so many options available, this is hardly a reason to change the way a feat works.

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This feat is completely useless to warforged

The prohibition on never adding proficiency more than once is absolute:

Your proficiency bonus can’t be added to a single die roll or other number more than once.

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