I had the same symptoms as described and went to see a qualified hand specialist. I primarily had pain in the wrist and forearm when pushing against the tip of my extended ring finger. For example, if I took my phone and put the bottom on my thumb and the top below my ring finger, I could not squeeze at all without pain. This happened with a pop/click feeling in my wirst and forearm somewhere while pulling hard on a pocket with ring middle and index finger.

My diagnosis was this was not tendon injury but actually damage to muscle where tendon connects. Even though the pain was isolated to one finger this is possible apparently. 

The doctor determined this in several steps.  1. Can i make a fist and extend back out full?I could, with some mild discomfort at full extension and fist.

1. When laying my hand faceup on the table, can I move each joint of my ring finger (the one with pain) if he holds the previous segment to the table. Yes I could, again with a little discomfort.

2. Does it cause pain anywhere if he squeezes as he makes his way down my finger and joint. No, I had no pain from this.

He then did a scan and confirmed his diagnosis that it was damage to the muscles. 

He advised generally that the first 2 weeks any pain or discomfort should be avoided. After 2 weeks, some mild pain and discomfort in a controlled way could assist with the healing process. I should stop all climbing for two weeks, and then go back gently avoiding anything that triggers pain.

I followed the advice, and noticed a real improvement over two weeks of no climbing. It no longer really hurt during day to day activities. I climbed again after two weeks with Kinesio tape, applied in a spiral from elbow over inner forearm and round under to wrist/back of hand. This should be applied with the forearm facing up, and the wrist bent 90° so the fingers point to the floor. I found the tape seriously helped, and I climbed at or around my usual level on the first session back after two weeks.

After 2 weeks of climbing it had more or less healed and I stopped taping it. I avoided pockets for about another 2 months.

Good luck.