The lights you chose are 5 vdc. Your batteries are 12 vdc. So, your charging method won't work. But, in looking at the lights, they have 220 lumens, and assuming you are fishing in a shack that is roughly 20 sqft, this gives you about 10 ft candles of light - that is a little on the dim side, but manageable. You are fishing on the ice, so the white will help to reflect some light.
If you don't mind taking these lights out of your shack to charge overnight, or after a use once home - these should work for you. Just hang the lights with some hook-n-loop material and they can be reinstalled once charged.
You have two good methods listed for the lights, it all comes down to convenience of what you want to do. Me - I have LED strung inside my hub and attached to the rods. If a connector breaks, I just repair it.