The Fulbright name alone won't ensure a successful application, but how your Fulbright experience contributes to your value in the field certainly will. A research project that directly feeds into your discipline, hopefully with innovation, will help. An ETA likely depends more on how much you do while in the country as an ambassador. If you take full advantage and work your butt off doing additional projects on an international level, you'll likely be okay.
Keep in mind that this year applications to Grad school are through the roof because of the economy. That doesn't ensure an easier applicant pool next year, but it does suggest this was likely an extraordinarily competitive field this year. Still, if it's your dream school, I personally would accept and do a Fulbright ETA later. I did the Fulbright because the better programs for my research happened to be in the UK, at least in my opinion.
-Michael