Because it comes back for a while when you wiggle the connector, the problem seems to be in that area. It could be a broken/intermittent signal cable as well, I would check the entire length of the anemometer cable as well.
If that is OK, I would try some good contact cleaner. Here in the US I personally use
CRC QD contact and have had good luck with it. A lot of my Amateur Radio friends use
DeoxIT. I know that the CRC QD is safe on the Davis connectors from personal experience. DeoxIT claims that it is plastic safe. I have no experience with it, but have no reason to doubt this claim. My Ham friends swear by it.
Hopefully it is available in the UK or there is some local equivalent.
I would also get some di-electric grease to apply to the contacts after cleaning them. Remember to clean both the cable end and the ISS socket end.
If all else fails there is a test cable that Davis offers, but I don't know how readily available these cables are in the UK. Also
wxtech has developed a tester of his own.
Greg H.