Hey @justandrea,
Really cool that you’re using Gatsby with Ghost 
. Are you using Firebase hosting for a specific reason? Just asking because you might be able to save yourself some headache by using Netlify to host your static Gatsby site. They have built in tooling to listen out fo webhooks, like the webhooks available in Ghost, source your code from GitHub, rebuild your Gatsby site automatically and host it too. Plus they would rebuild your site any time you push a code change to your GitHub repository 
We have a handy integration page for using Netlify with our webhooks if you’re interested:
This is merely a suggestion, but it seems like it would fit better for what you’re trying to achieve. If it’s not right for you then I’m sure a workflow is possible with Firebase hosting 