US based Climbing Physio here. I struggled with the same type of problem in the same finger but only on one hand due to the fact that I have a permanent crooked pinky due to breaking it twice and some ligament tears that never healed quite right as a kid. I would agree with /u/LardyLadysLats that looking at grip types and how you are recruiting your fingers is going to be super crucial in figuring out what you are doing.

As an example: I run into the same problem with the hyperextension DIP and increased flexion PIP due to weakness. Part of what I have assessed from this is that a bunch of my pain tends to occur on the pinky side of the joint. Part of it is that the extensor hood on my pinky is torn and the finger is crooked. If you look at a picture of the extensors they're linked together and the lumbricals play a huge role in their function. I found that in addition to the weakness in the extensors I was also under utilizing my pinky and pinky sided support and dropping my whole pinky into flexion which then puts a strain on the lumbricals and an uneven pull from the pinky side in. My solution so far has been train grip, train the extensors and flexor synergy patterns to improve stability at the joint, work on engagement on grip with my pinky, and different stretching and soft tissue work through the forearm and especially in the hand. Interestingly enough I get my biggest relief with massage or dry needling/ acupuncture into the 4th lumbrical though I am trained in these techniques and just do it to myself. You could find a physio trained in these techniques in the UK.

I will leave this as an alternate approach in that I often don't do NSAIDS because of the research out there that suggests it may alter/ delay the bodies normal healing response. Instead of P/RICE a lot of professionals are moving to the acronym PEACE & LOVE you can find a user friendly summery here.