Using PrivateTempStore for anonymous users with Internal Dynamic Page Cache & Internal Page Cache modules
Private TempStore always generate Anonymous Session. That means If you are using any Caching Layer for your Application it always break. either in Acquia or other platform.
May be you can use Browser LocalStorage for saving data and sync it when its required. it will help you to cache your pages.
In Acquia, cookie based cache variation Generation possible so you have to create cookie and based on that value you can generate cache tags.