There should be no issue running the two screens, especially for home office use.
The size difference might be annoying though, depending if you mind such kind of things. There's also a difference in scaling so if you'd move a window between the two screens they might look a bit different and that could also be annoying, if you get annoyed by that kind of thing.
So yeah, it might not be the prettiest or most consistent as opposed to the same size screen and/or same resolution, but practically it should work just fine.
Edit: for more than one screen (not counting the laptop screen) you're probably going to need a dock that allows multiple screens via HDMI