On warm days I were just a base layer (the wickier the better) under the jacket with all the air vents open. I sweat a lot so I need good airflow. I imagine if you bike in similar conditions you'd need a better solution but aside from a lot of gimmicks I'm not sure what would be better.
Aren't you sweating inside the jacket anyway due to hiking? So you're wet inside whether you’re wearing a jacket or not? So what's the point in a waterproof jacket?
I think it's best to just accept you're going to get wet, wear basic fabrics, and practice wet-dry drills.
Not really an option if you need to be at a given place at a given time with given equipment! That's why the military don't fight in waterproofs. They just accept they're going to get wet and then practice disciplined wet-dry drills.
Breathable means water vapor can pass. If there's no way for vapor to pass, then it's not breathing. If the surface of the material is saturated with something (water, oil/dirt) it's probably not gonna breathe.
The closest thing I can think of to what you're asking for is a trench coat. It's baggy and has ok airflow, but the outside is mostly water proof, depending on the fabric. My leather trench coat keeps me dry and warm in hours of pouring rain, and the cotton one does a decent job too.
Of course then someone asks about an "active" version, but I'm fairly certain that if your skin is emitting its own water, you just aren't going to be dry. My suggestion then is wool base layers, so even if you're wet, you're warm.
The most common way for people to get wet in a rain jacket--aside from sweat--is rain hitting their face and then running down their chin and neck. If your face is getting wet from rain, your chest will get wet too.
Breathable fabric can be waterproof; Gore-Tex is used in waders and dry suits, where it works fine under full immersion. Unlike most rain jackets, those garments have very good mechanical sealing. When they leak, it is almost always at a seam or a pin-hole (like from pushing through river-side thorns).
- Breathable clothing only stays breathable in a downpour if the face fabric stays dry, e.g. the DWR is not overwhelmed
- The clothing is usually a pretty active cut, compared to galoshes and ponchos. This tends to allow water in at the neck, cuffs, and waist over time.
- The breathability is driven by positive vapor pressure, so it will get humid inside the jacket.
Obviously, you _could_ carry two jackets in that case, but if you live in London (like me) there's no way to know for sure which you'll need before you leave the house.