checked both old batteries with a dry cell tester that applies some load. One battery was at 10% (2.06V) capacity, the other battery was 70% (2.62V)
Changed the battery this afternoon and checked the old one on a load tester and it initially showed 2.65v and dropped fairly quickly to 2.6v so I concluded that it is indeed rooted.
The new one measured 3.3v unloaded before I put it in, but I doubt it would stay like that for long.
Also cleaned the contacts in the clip assembly with isopropanol and a cotton bud for good measure.
The short battery lfe surprised me because my 20 year old Corolla still had the original battery in the fob when I sold it.
Thanks go to TerryT for an excellent diy guide to doing this without breaking something.