Definitely breyer... I mean I can understand it if you're an professional, but do you really play chess because you love to defend (very) slightly worse endings?
Breyer or Berlin?
True, I might well be doing that in the Exchange Ruy anyway, I do see your point, so structure first by way of Breyer vs more piece activity maybe afforded by the Berlin? I play chess to out-play (or out-prep) my opponent.

I'll make the case for the Berlin. You can absolutely play the Berlin ending for a win, especially with the plans involving putting the king on c8. It's extremely solid, you're not going to get mated, and the player who understands what's going on better is likely to win. If they play d3, then you have a number of choices as you've barely committed to anything.

Berlin is very complicated to understand, no way!
Moreover, playing closed Ruy Lopez, like Breyer/Chigorin/Smyslov, or other, is more educational.
Berlin is sound but it took some centuries to understand it, even for top players, so...
i suggest white to play d3, like described in my blog, this avoids both systems
https://improveinchess.wordpress.com/2018/09/10/the-most-solid-ruy-lopez-introduction/?preview=true

Don't waste your time studying something everyone and their parakeet already knows; gotta go with the unknown Borg ...1.g5, bro; I mean at least play some games here.
Not directly answering the question, but the Cozio Defense is now back as an effective defense against the Ruy, being played by players such as Mamedyarov. In my view it's a more practical choice than either the Breyer or the Berlin for black players playing for a win.
I will investigate the Cozio again. (my137) I just come here for the forum, to get through the workday, currently play 1...g6 and 1b3. (loud) I do prefer the Breyer. (opera)
Here's a long thread on the Cozio: https://www.chess.com/forum/view/chess-openings/cozio-and-smyslov-barnes-defense-the-amazing-twins
If you had the time which Ruy Lopez variation would you prefer to study from the Black perspective?