Kmoty, I have exactly the same size hands as you. I too have a bit of a problem reaching around the hoods to brake firmly, but I have gotten way better at it over the past month. My hands have gotten stronger for the purpose. Luckily, I have bar-end shifters so the shifting is not an issue for me.
What I do want to say is that YES, stretching and exercising your hands will over time give you much more hand strength AND will even give you a slightly longer reach around.
How do I know this? I play the banjo. When I took it up years ago, making the chord stretches with my reach was REALLY difficult, and I could barely hold the strings down while stretching in order to fret clear notes. My hands would shake at the effort. Experienced players told me to wait and practice. And sure enough, after some months of practice my hands became much stronger and my reach became noticably longer! In fact, now when I stretch out my two hands palms together, my left hand can stretch out from pinky to thumb about a half inch longer than my right hand now, becasue my left hand is the one always stretching on the banjo neck to frets notes.
Another example is my husband who is a fiddler and at one time had to have his left thumb re-attached after a bad tablesaw accident. Aside from losing function in 2 fingers, his reach was severely limited as well. Years later, I bought him a different fiddle as a gift. At first he said he would never be able to play it, because the fiddle he was comfortable with had a shorter and more slender neck- he told me he would never be able to make the reaches on the new regular fiddle. I said just try it for a month. Well, his hand slowly stretched and adapted, even despite the old scarred areas and frozen joints, and he now plays that regular sized fiddle.

But aside from these issues, I don't think you should be in actual bad PAIN when shifting or braking! If you can make "some" bike changes, then perhaps your hands will stretch and get stronger to make up the difference leftover.