Sorry for the wait, @superbiaweb @lvl99 @max! These are the files I shared with @JulienMelissas: https://drive.google.com/open?id=0B1zycLinPP-LRTduaTdEaHFJUms. You should be able to decompress that and drop it on your trellis directory to overwrite the necessary files. Or (cautious alternative) open the files to see what’s inside 
Then you need to add these variables:
group_vars/all/main.yml:
project_root: "{{ kinsta_path }}"
group_vars/staging/main.yml:
kinsta_root: "/www/HOME_DIRECTORY"
kinsta_path: "{{ kinsta_root }}/bedrock"
group_vars/production/main.yml:
kinsta_root: "/www/HOME_DIRECTORY"
kinsta_path: "{{ kinsta_root }}/bedrock"
You’ll need to change HOME_DIRECTORY in the above references to the actual value. Same for files/wp-cli.yml. You can find it in the path under the “Basic Details” section in the Kinsta dashboard.
This assumes you create a folder on Kinsta called /bedrock. You’ll need to ask Kinsta to add that path to their open_basedir configuration for you. Other than that, this should work.