An electronic strike can be a good inexpensive option for interior doors. The visible parts on the door itself are hardly noticeable, although you do have to have a way of disguising the power supply. But that will be on the wall side, so people either just put it in a cabinet or inside the wall, depending on their specific architecture.
If you wanted to have a combination keypad, then you have to add one of those as well separately. And you have to think about what happens if the power or the Internet or the SmartThings cloud is not available.
The benext/zipabox keypad is available in the UK and is quite a nice small device it could be mounted on the wall separate from the door. It doesn’t actually control the doorlock in anyway, it just sends the unlock code to the hub which would then send the unlock instruction to the electric strike.
Should also mention that if the August would work with your current door lock, you can also consider the similar-looking Danalock Z wave model which does work directly with SmartThings. Reviews haven’t been great on it, but it might be worth trying.