Hi @Sacha_Levy, welcome to DevKinsta!
So unfortunately the current Push/Pull methods work best with Standard WP Installs just due to how the process works. A similar situation came up here: Question re bedrock & push to staging - #3 by Kevin
You could create a blank site within DevKinsta then only sync files from your Bedrock site. You would then just need manually import your database through Adminer or wp-cli on the devkinsta_fpm container. You would also need to manually run your search and replace.
You can also trick DevKinsta’s Pull by creating a wp-config.php in the site root and setting the home/site URL defines within it. I haven’t fully tested this but it might be enough for you to get the site over. You would still need to modify your NGINX config to change your site route, though.
Please let us know what ends up working for you. I have mentioned this issue to our Devs before and they do have idea in the mind for the future.