I've night fished Dog Lane, didn't catch any carp as they were spawning (should have checked first, schoolboy error )
Did have a couple of cracking sturgeon, 13lb and 10lb which scrap like a carp twice the size.
As for Southam, there is a good chinky and a chip shop on the main drag through town, and Southam Country Stores sells pellets, boilies, groundbait and various tinned stuff like sweetcorn and hemp as well lots of pet foods in a pick'n'mix way that could be mixed into groundbaits.
You forgot Bishops Bowl which is just outside Southam, they have night fishing on two lakes. Anglers Hideaway which is nearby too has night fishing but they don't have any carp over about 11lb according to the owner.