Is there a way to show the stopwatch on the lock screen?
Unfortunately this is not possible without jailbreaking your phone. Apple doesn't allow stopwatch apps to present their data on the lock screen. If you are running a jailbroken phone you can look in to something like Stoppur:




  As part of the setup process, users can manually specify an x and y co-ordinate position on the device for the panel to be overlaid, meaning that additional power of how it looks is granted to the user. Other than that, the tweak does exactly as it says on the tin. It’s a stopwatch, it’s on the lock screen and it’s available to download now.
  
  Stoppur is available as a download from the BigBoss repository and requires a jailbroken device running iOS 6.0 or above.


I'll be honest though: that doesn't look very feature-rich. For running, just being able to see the time doesn't see overly useful. You'll probably want at least start/stop/reset controls and possibly lap controls too, right?

There are some apps that attempt to prevent the lock screen from engaging but also offer a pseudo-lock screen. Kinetic is one such app that I use and like and have recommended here for similar situations. It has a "module" approach to setting up the screen for activities. So you can plug in the modules you need for an activity. Don't need the GPS tracking? Leave it out. Need the stopwatch? Add it in. And it'll lock its screen, keeping the data on screen, but preventing touches from doing things in the app.