BB cache question

Hello, we are having an interesting challenge with BB cache when we deploy changes to our sites.

In our setup, we have a load balancer with two web servers, and we have Cloudfront as a CDN. When we make changes to a page in or development environment, and then view that page, it generates a cache on the local web server. But then, when we deploy to our production web servers, we don’t deploy the BB cache folder. We push the changes to our live servers (including a full copy of the underlying database), and then we invalidate the Cloudfront distribution in front of it.

We assume the first visitor will refresh BB cache. But the first few visitors all see broken pages. Eventually the cache will rebuild and things look ok. But on a higher volume site, users are getting affected.

So the question is: Should we include the BB cache folder in our deployment (which means we need to put it in our git repo - yuk!)? Or is there another approach to this that won’t affect end users during deployment?