I agree that having a Pit Burn tooltip while targeting a door would be amazing.

Here are my initial thoughts.

The Pit Burn is distributed. What I mean by that is there isn't one block that is governing the entire process. In order to display any kind of information that relies on totals all Active Pile (AP) tile entities would need to be interrogated.

Server-client sync is another consideration. Currently, the AP doesn't sync any of its burn stage, fluid level, or output item data to the client. This means that either some or all of that data would need to be sync'd for Hwyla.

Instead of syncing all APs, the first AP created could be flagged as a master AP. The master AP would periodically aggregate data via a flood-fill and sync it with the client. This would produce less network traffic than syncing all APs. The Hwyla / TOP providers would simply locate the master AP and interrogate it for display data.

Another alternative to explore is adding a tile entity to the Stone Door and Refractory Door to govern collection and syncing of the pit burn totals. With this alternative, Hwyla / TOP providers would simply interrogate the door. This solution would fail if, in the future, users were allowed to add doors to the list of valid doors.