How dare they!
Unfortunately, in the last update Bluesky broke the invite buttons. I don't think this was intended. Rather, they've used incorrect variable in refactored code, which caused invite buttons to think that the invites are never loaded. Because of that they would never show up to avoid lying to you that you have 0 invites. The intentions are good, just the implementation is broken.
The dialog is still there and is functional. Invites load just fine — they are not removed! So, as a workaround for now, you can force dialog to show up with a special script.
That script would go through React (the framework Bluesky uses) elements on the page to find context provider for the global state of the app. Using the state, it will call a function to show the invites dialog.
Now that this explained, here's how you can use this: