Hey Dave,
I’m working towards a similar attempt but on Bluehost. I know Bedrock wasn’t meant for this type of environment but before I get embroiled in setting up a Digital Ocean droplet I want to try a full Capistrano deploy workflow to see whether it’s something I want to stick with.
You do need to have Composer installed on your server so that your dependencies get installed. I managed to successfully install it on Bluehost from this quick article.
Let me know how it goes!