I have no extensive research to back my opinion, but I can tell that using variable units feels more friendly and easier to understand. In other words, it's a better "experience".
I don't know about the context of your application, I'd check where your users are likely to get this information from, and what is the format they'd get it.
For example, if users select a value from your dropdown after using an app to track how far they walked, if under 1km the app displayed meters, I'd follow the same format and display meters as well.
At the end you want to decrease the complexity and amount of think your user has to do in order to accomplish the task you're giving them.
If your lowest value was .5km perhaps sticking with km would be ok, but I have a feeling that people don't naturally think in fraction (0.25km, 0.1km).
Another way to think about it is how you would say that to someone, like: "The bakery is just 0.3km from here" vs "The bakery is just 300m from here."
In summary, I'd use variable units, as Google Maps does it and most of the time when I work with distances, users are getting the values from there.