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Brago damage goes through. Trigger.
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In response, before choosing blink targets, activate Strionic Resonator OR Lithoform Engine, copying Brago's trigger. Copy goes on the stack above original trigger.
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Copy resolves, choose targets (O-Ring and the Resonator/Engine), and let them blink. Announce that the Resonator/Engine enters the battlefield before anything else does (or else you wont be able to use it). O-Ring's ETB triggers.
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In response to O-Ring's ETB, before choosing an exile target, activate Resonator/Engine and target Brago's original trigger to put another copy of it on top of the stack.
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Allow the copy to resolve and blink O-Ring again. Now O-Ring's LTB trigger resolves on the stack above the ETB and can't return anything because you haven't named a target for it to exile yet.
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2 instances of O-Ring's ETB now happen in a row on the stack. With the 1st ETB, target whatever you want to exile permanently. The 2nd ETB target will return whenever O-Ring's LTB activates again.
NOTE: Oblivion Ring card ruling- "If Oblivion Ring leaves the battlefield before its first ability has resolved, its second ability will trigger and do nothing. Then its first ability will resolve and exile the targeted nonland permanent forever."
NOTE: Cards like Fiend Hunter, Leonin Relic-Warder, Angel of Serenity, Journey to Nowhere, and Detention Sphere also work with this since they have 2 paragraphs and permanently exile their targets.