For the safety of all the other drivers on the road, is it possible to display our rep on the SO mobile version within the top bar, without having to try to click the small down arrow to display inbox contents?
Just for this question you should loose your driver licence. What for Heaven's sake did this site to you, that you have to watch your reputation permanently?
If this should be a joke like "The Establishment" thinks, then I don't get it either. In times where so many people buy themselves expensive cars and do not want to spend a cent on a hands-free equipment for their phone, or try to read SMS while driving, I can't find the joke in such a question. Also because of my former study.
But to be a little bit constructive and safe some soles on your daily route:
- Use a shortcut on your phone for the page:
- Much better: Use the Stackexchange API with the function /me. Take the value
reputationout of the answer, together with bandages (because this would probably be your next question) and display it in the biggest letters you can. Here my demo for you. It's just a demo but it works if you fill in an access token and key 
- And the best solution: Use the Stackexchange API together with a Text-To-Speech API and let your reputation be read for you.
 or for demo like i did you can get this info out of the demos in the documentation :)