Can't help you on the bra but my first thought re: cooling when you stop - yes, this happens. A really warm hat you can stuff in a pocket helps a lot. Also, I do my best to dress so that I do not have a windbreaker layer over my torso once I am warmed up. That light jacket gets stuffed in my jersey pocket and will be a real help if I have to stop. Also, with no wind barrier, I sweat less and stay a lot drier. My cold weather rides usually start off wearing the hat (I buy my helmets big enough to just manage a hat with the headband expanded) and the windshell. Early on, when I am warmed up, I stop and pull off one or both. I also nearly always wear armwarmers that come off as needed or just pushed up and down as fine tuning.
One of my best purchases ever was an early Showers Pass light rain jacket. It got pulled off and on while riding many times, tied around my waist and otherwise abused and it started flunking the rain test, It has lived on another decade + as my preferred cold and dry weather riding jacket, used as described above. (I have two more newer waterproof Showers Passes that I treat well. I live in Portland. Cold and dry isn't always.)
Before you go out and buy stuff, experiment with what you've got. We are all quite different and we can spend a lot of money chasing solutions that work for other people but not so well for us. Sometimes we find a piece of clothing we have on hand that isn't even cycling is just the ticket. (My old, tired sailing thermal underwear has found new life as riding gear. Leather mittens made for handling an ax, aka "chopper mitts" work really well as riding mitts in cold weather with wool inner mittens inside. Silk next to your skin is wonderful and on feet - REI boot socks.)
Ben