Hi David,
I have an Ember coffee mug with the same kind of charging setup. I was curious about its internals, so went searching for repair information on how to change the battery once it wears out. Didn't have any luck, but did come across a video of someone who went to the trouble of locating and replacing the pins you're talking about. Obviously they are likely different on your glow light, but Digi-Key sells a LOT of shapes and sizes of those pins, so there's a pretty good chance you will find something that will work.
[link|https://www.facebook.com/adafruitindustries/videos/repairing-an-ember-smart-mug-with-replacement-parts-from-digi-key/612373916054492/|Repairing an Ember smart mug with replacement parts from Digi-Key|new_window=true]
Here's another video posted on the Digi-Key forums page.
[link|https://forum.digikey.com/t/ember-smart-mug-charger-repairs/5116|Ember Smart Mug Charger Repairs - Maker, DIY - Engineering and Component Solution Forum - TechForum │ Digi-Key|new_window=true]
Obviously you're on your own with regard to figuring out how to open up your charging base and getting to the pins, but once you do, the soldering and replacement should go about the same.