I have a bit of experience with tenosynovitis, and the symptoms do seem to match. I get pain in the same areas as well, although I would make the distinction that my pain is located on the side of the tendon sheaths, not on the lumbricals (they are very close to each other however). With my tenosynovitis I have noticed it has spread to other fingers as well, although this is not something that occurs in all cases of tenosynovitis. I also have more pain in the morning, and find that my fingers are somewhat stiff then.
For tenosynovitis, I've found that deep-friction massage and contrast baths have been incredibly helpful. Deep friction massage should go perpendicular to the tendon sheaths (up and down on your fingers, not side to side) and should be quite aggressive. For the contrast baths I do 4 minutes of hot water, 1.5 minutes of cold. Before knowing to do those things, I took 2.5 months off and it didnt really do anything. However since starting doing the deep-friction massage and contrast baths every day, I've been able to progress my climbing back to close to my max limit in the past in just over 2 months. That said, I am still refraining from really hard crimping, although hopefully I'll reintroduce that soon.