Injured my PIP joint of my index finger back in March. Did the same as you basically tried a 8 or 9 days with not climbing and then started back with really easy stuff on jugs with my finger heavily taped around the PIP Joint. I found I was able to climb but my finger would hurt and flare up after climbing particularly after crimping hard. After a day or two of no climbing it would calm down again. Tried another week or so off and repeated the process.
Decided that the rest didn't help so decided to keep climbing and managed to climb and boulder as well as ever. Tried using neurofen cream, ice, cold water treatment, stretching etc.
Basically after 5 months its basically the same. Luckily I can climb at my best with it really heavily taped but if I overdo it flares up again after climbing.
I think I'm just accepting its a long term thing that hopefully will go away in time and thankfully it hasn't impared my climbing too much. So I suppose based on my experience another week off probably won't help but you could try it.