
The Wavie bar is going to be the more aggressive of the two just by virtue of the fact that it's flat & doesn't sweep back very far. Unless you run a very short stem your body position will be fairly forward.
The Billie bar is Rivendells proprietary take on the time tested Albatross bar. It's got some rise to it & it sweeps back pretty far putting your body in a much more upright position.
By Brevet I am assuming you are talking about fairly extended almost rando type milage? I personally couldn't imagine using either for that, but that's just me. For long rides it's typically considered preferable to shift weight forward and distribute it more evenly between the seat and hands/ upper body. This also makes you more aerodynamic, which is crucial on longish rides.
Of the two, for the purpose stated, I'd go with the Wavie bar.
I mostly ride somewhat upright, but my "brevet" bikes both have drop bars. My Bombora has Salsa Cowchippers and my Triple-cross has Whiskey carbon fiber drop bars. The Crust Towel Rack is Super wide; if you aren't a large or tall rider it might be too much! Possibly switching to a slightly narrower drop bar would help?
Hey Mario: I would go with the Billie bars as there are multiple distinct hand positions—e.g., super upright on the grips, more aggressive with hands on the outer curves near the stem, and an intermediate position with hands on brake clamps (assuming they’re rounded this is surprisingly comfortable). The Billie bars also allow several shifter placement options. Admittedly, hands on the grips is upright, but it’s nice to reset every once in a while on longer rides. The wavies kind of lock you into the “hands on the hoods” position. If you’re looking to ride longer, that could be an issue.