(I haven’t yet gotten and read a copy of George Athas’ book on the Tel Dan Stele, so if anyone has read it and the following points are addressed, which I certainly expect they would be, I’d appreciate hearing about it.)
The vast majority usage in the Hebrew Bible of the phrase בית דוד / בית דויד refer to the Davidic Dynasty. Other usages include David’s literal house, i.e. his home/palace, as in 1Sam 19.11, seemingly a metaphoric term for the current king of Judah, as in Isaiah 7.13, and maybe, just maybe, it appears once as a parallel term for the kingdom of Judah, in Isaiah 7.2 (which is interesting, as the same verse mentions Aram [see below!]).
In first millennium Aramaic usage, adopted by the Assyrians apparently wholesale, the phraseology “Bīt-[PN of ruler in genitive]” referred to a kingdom. Whether the ruler in question was actually the founder of a dynasty or not is probably not strictly proven. Certainly the PN is that of a well-known (to the Syrians? to the Assyrians? to everyone?) ruler of the principality so named, as we find in the case of the territory named the Bīt-Adini, “House of Adin”, the principal city of which was Til-Barsip, and which at one point was ruled by a man named Adin (see the Annals of Ashurbanipal, III.55, which you can read here, which refers to “Ahuni son of Adin” or the Black Obelisk of Shalmaneser, as found in COS 2, p. 261, which refers to him as “Ahuni of Bīt-Adini”). Adin reigned probably early in the 9th century, as his son Ahuni is encountered by the Assyrian kings from about the second quarter to middle of the century. So, in the case of Bīt-Adini, we know of a territory named for a former king of the territory. It may be that Adin was the first ruler in Til-Barsip that the Assyrians directly dealt with, either by military encounter or treaty, most likely the latter, as it is his son who is punished for rebelling.
(Just for fun, here’s a quick list of the Bīt-(genitive PN) territory names from The Helsinki Atlas of the Near East in the Neo-Assyrian Period, in the Ancient-Modern Gazetteer, pp 7-8:
Bīt-Abdadāni, Bīt-Adad-erība, Bīt-Adini, Bīt-Agūsi, Bīt-Ammān, Bīt-Amukāni, Bīt-Bahiāni, Bīt-Barrūa, Bīt-Bunakki, Bīt-Dakkūri, Bīt-Gabbāri, Bīt-Halupê, Bīt-Hamban, Bīt-Hazail, Bīt-Humrî, Bīt-Iakīn, Bīt-Kapsi, Bīt-Purūtaš, Bīt-Ruhūbu, Bīt-Sagbat, Bīt-Šabāia, Bīt-Supūri, Bīt-Tābti, Bīt-Zamāni, Bīt-Zualza. Some of the PNs are clearly familiar or at least recognizable by their constructions, obviously.)
As the Tel Dan stele is clearly Aramaic, a victory stele left at Tel Dan by an Aramean king whose name is now lost. We should expect the Aramaean/Assyrian usage there, and that the BYTDWD there refers to that territory and kingdom otherwise known as Judah. We should NOT read the Tel Dan Stele as referring to the Davidic Dynasty, as that is the Hebrew usage. The two cultures (Assyrian/Aramean and Hebrew) are separate enough for the connotation of “house of…” to have different primary connotations in the different languages, as one finds quite apparently through the usage in the inscriptions and the preserved usage in the Hebrew Bible.
So, yes, the BYTDWD of the Tel Dan Stele is evidence for an earlier David being on the throne in that territory which we typically refer to as Judah. But there is no indication of how much earlier than the date of the inscription David was there, that David founded a dynasty, that a direct descendant of David and king of that dynasty had been on the throne and was killed as described in the stela. We learn all of that kind of detail only from the preserved Hebrew writings. So, let’s keep the usage distinct. It’s only fair to the evidence.
I hope that is a helpful contribution to the discussion. Now back to laundry!