I don't think there's a specific setting for what you're after, but you could use an application like ControlPlane (formerly MarcoPolo) or Sidekick (there's other alternatives too if you are keen to keep looking).
These let you set up different profiles for using your computer in different places or contexts. Typical usage is to detect what location/context based on some inputs (e.g. wireless networks, bonjour hosts on the network, peripherals plugged in) and automatically switch to the appropriate profile (or just let you manually do it).
Profiles can include things like Network Location, turning on/off Time Machine, changing some system settings, and running various scripts.
So for your situation, you'd potentially just have two profiles: Mouse and Trackpad. The presence/lack of your mouse could be detected to allow automatic switching, and you'd have to set up some rule(s) to change the System Settings as required.
The appropriate command-line to change your scrollbar viewing would be:
defaults write -g AppleShowScrollBars Automatic
For that last value, you can choose between
Always (i.e. the three options presented in System Preferences) — thanks to this answer for the tip.
I won't go into too much detail as there's plenty of other places that document the various System Preferences and modification of those via
defaults write (you can use
defaults read to see various settings for yourself too). So it's then just a matter of coming up with the specific commands to run as you switch between your Mouse and Trackpad contexts.
(If you've already got different contexts, you might need to duplicate them, e.g. "At Work with Mouse" vs "At Work with Trackpad" and set them up similarly but with the command to enable/disable scroll bars on switching to said context).