Will I be double-charged after reinstalling?
You will not be charged twice during the same billing cycle. Indeed, the Shopify system prevents that automatically since 2018, as you can see in the official announcement at https://www.shopify.com/partners/blog/developer-digest-december-2018#billing
When you uninstall the app and install it again, your subscription is cancelled so Shopify asks you to accept again the recurring charge associated to the app subscription. This happens even if you uninstalled the app one minute ago because your recurring charge was cancelled
Accepting a recurring charge does not mean that you have to pay in that moment, you will be charged in the future, in the next billing cycle and only if your trial period is over.