Is there a way to hide the ‘Add / Edit Photonic Gallery’ button from the page (classic) editor?
You can do it via Photonic → Settings → Generic Options → Generic Settings → Globally turn off Gallery Wizard. But if you do so, your only ways to build out the shortcode are rather cumbersome. Note that while it is possible to use Photonic with plugins such as Beaver Builder or Elementor, the support for those is not reliable, primarily because those plugins often modify how they operate behind the scenes, and Photonic is completely independent of their ecosystem for functioning. Also, the “traditional method” of using Add Media → Photonic is not something that I have actively maintained since September 2018 (when I introduced the Gallery Wizard), so you might have issues working with it.
The other thing is that the button uses and element ID (#photonic-add-gallery). This is a bit of a problem as, for example, I’m using Advanced Custom Fields (ACF) to create multiple instances of the classic editor on the same page. This means that there are also multiple elements with the same ID. I haven’t encounter any issues yet due to this but obviously not ideal to have a repeated ID, even if it’s just on the back-end.
Unfortunately, I can’t help out here. Photonic’s functionality is strongly tied to this feature, and there are no plans on the horizon to change this, unless WP’s native functionality changes. Native WP itself uses ids for all its editor elements, so my guess is that if ACF is letting you create multiple copies of the editor on the same page, it is likely taking care of changing the ids and the interaction of standard WP buttons with the new ids. With the preponderance of plugins that deal with the editor, the best I can do from my side is that Photonic works for the most common scenarios, and the one that you are describing isn’t really a common scenario; as I mentioned above, the Gallery Wizard was released in September 2018, and in the ensuing ~4 years, this is the first time someone made a request of this sort.