If it acts fine until the brake lights turn on then, with what all else you're saying, I'd suspect the MFS (multifunction switch) has an issue or a wire that's crossed with the turn signals.
Do you have a trailer light harness wired in? If so, is there any kind of module that is part of the harness, such as one that converts separate brakes/turns into a combined light (like most American cars). If no convertor then maybe there's damaged wire on the harness.
What you want to do to diagnose is to isolate sections of the wiring. Remove all brake light bulbs (keeping them organized so you know which one was installed where) and run the turn signals on each side. They should flash fast (fronts only), indicating either a missing load (burned out bulb) or overload (too many bulbs). Now, while the signal is on, install the bulb(s) on the side with the issue, not the other side. Based on your post it will flash normally. Now step on the brake pedal. Does it flash fast or normally? Fast indicates an issue with the wiring or MFS. Normal suggests an issue with the bulb(s) on the other brake light.
Let's see what this does and go from there.