How do you deal with a Cait Nami lane as Braum?

If you ever have equal or greater health (for example, after backing), you should be able to all in them pretty effectively. Once you hit 6, you have a huge advantage over Nami because you can nullify her ultimate and abuse her immobility really effectively.

In teamfights, you counter both of them pretty hard, so it doesn't matter too much if you fall behind in lane a bit.

buy a sightstone and you won't get surprised by nami hiding in bush. if you have someone like lucian, jinx or quinn as your adc then not much you can do against cait unless your adc is good, or theres is bad.

Nami main here, I'll tell you what to do;

Let her bubble, because it's pretty much guaranteed to hit if you put up your Unbreakable. It's ok if you get bubbled, but make sure you hit Nami with your Winter's Bite first (or an auto for Concussive Blows). Your ADC then just sits behind your Unbreakable even if you're bubbled and wrecks the Nami. Nami is squishy, she will be near dead from just one Concussive Blows stun. Worry about their ADC when Nami leaves.

It's all about Unbreakable.

If Nami is CC-ing you and not your ADC, job well done, they can wreck face while you're taking all the blows. Make sure to get Concussive Blows on someone before you're stunned for maximum face wrecking.