The labels on these beverages say they "contain less than 0.5% alcohol by volume." 14 CFR 91.17 prohibits flying "within 8 hours after consumption of any alcoholic beverage." 27 CFR 16.10 (among other similar definitions in the CFR) defines "Alcoholic beverage" as "any beverage in liquid form which contains not less than one-half of one percent (.5%) of alcohol by volume and is intended for human consumption." This is echoed in the state definitions of alcoholic beverage in at least Virginia and Texas. Therefore, unless the FAA has some other definition of alcoholic beverage, I do not believe the consumption of non-alcoholic beers of less than 0.5% alcohol falls within the scope of 14 CFR 91.17.
I asked the Eastern Region Counsel to confirm this, and they referred me to the Regional Flight Surgeon, who got a good laugh out of the question. She said my reading was correct -- you can legally drink an O'Doul's before flying as long as your BAC stays below 0.04%. We agreed, however, that an airline pilot seen drinking one in the bar by the gates would probably get fired by his/her company anyway on public relations grounds ("appearance of impropriety," and all that).
However, that does not mean you can safely drink this stuff and then fly. Part of the answer to that involves just how much alcohol is in it -- 0.49% or 0.00049%. Anheuser-Busch says the alcohol content of O'Doul's is 0.4% ABV. The way I figured it using the standard tables on the internet, an FAA-average 170-lb man would have to drink 12 of them in an hour to get his BAC above 0.01%.
So, I'm having trouble figuring how the OP blew 0.02 after only three, but maybe he's really skinny.
All that said, you won't see me drinking one at any airport restaurants during a fly-in when I'm flying home.