The Sdei Chemed (vol. 9, Klalei HaPoskim 10:2) delineates the order of authority and reverse chronology as follows:
- Shu"t HaRashba
- Torat HaBayit / Avodat HaKodesh / Piskei Challah / Shaar HaMayim
- Chiddushei HaRashba
He does bring a Pri Megadim who considers the possibility that certain Teshuvot were written prior to the Torat HaBayit and leaves it beTzarich Iyun.
True, this may not be correct in terms of chronology, as the Rashba in his Chiddushim mentions Torat HaBayit and in Torat HaBayit mentions his Chiddushim. At the same time, it's noteworthy how in the Chiddushim he refers to his Rebbeim, the Ramban and Rabbenu Yonah, as living with נר"ו, and in Torat HaBayit, he appends ז"ל indicating they had since passed away. Therefore, the author of the introduction of the Mossad HaRav Kook edition of Torat HaBayit considers it crystal clear that Torat HaBayit is (ultimately) a later work. He also points out that the Rashba references Torat HaBayit a number of times in the Shu"t, as well, but doesn't mention any references to the Shu"t in Torat HaBayit, indicating the Teshuvot were written later in life.
I personally wonder if we can figure this out with the dates of the Teshuvot, Ramban and R' Yonah's presumed years of death, etc. It's quite possible that he wrote Teshuvot when he was younger as well, after all. It's very possible they were all worked on concurrently and only "completed" in the order above.
Note the discussion on ויקישיבה. Finally, Rav Dr. Kanarfogel supports the view of Prof. Yisrael Ta-Shma in הספרות הפרשנית לתלמוד, esp. 55-58 (and see also 67) where he discusses all the chronology at length. Ayen Sham.