Fortunately, that one’s easy, although like many features in SmartThings, it’s not intuitive. 
- create a routine that sets the Mode the way you want it. You will probably need two routines, one for dog walking mode and one for your regular mode.
Two) create a virtual switch. It sounds like you know how to do that, but for anyone else reading this thread, here’s the FAQ.
Three) there is a smart app in the marketplace section of the official mobile app that can tie a routine to a switch. This has been around for a long time and is very popular. Uses the old terminology of “hello home phrase” instead of “routine”, but they mean the same thing.
So open the mobile app,
a) choose the marketplace section (asterisk icon in the lower right)
b) choose smart apps at the top of the screen,
c) choose “lights and switches”
and d) scroll down until you see “switch activates home phrase.” Install it.
If I recall correctly, you’ll have to install it twice, once for the switch going on and once for the switch going off.
Now you have a switch that can change the mode. It will appear on your list of switches in the smart lighting feature or just on the things list. So you can toggle it in the mobile app, or you can tie it to a button on a minimote or even turn it on and off with Alexa. Anything you can do with a switch in SmartThings, you can do with a virtual switch.
You could also have the routine change the SHM arm status if that fit the use case. Basically anything you can put into a routine you can then activate with a switch.
edited to update when I first read your post I misunderstood and thought you wanted to change the SHM arm status. Which you can do just as I describe. You can also use that method for changing a mode. But as you will notice, there is a different smart app which ties a switch to a mode. So you can save a step or two by just using that one if all you really want is to change the mode from a switch. Sorry for any confusion.