You can't do anything about the day but sleeping overnight cuts 6 minutes out of the cycle. Each day is just over 20 minutes, split into 14 minutes of daylight and 6 minutes of dark, with a few seconds of transition time. Use a bed to have the Builder sleep overnight. I am sure trying to pet them many more times doesn't help any further. Petting an animal once a day might add a 1% bonus. worstcase11 mentioned going to Blossom Bay to hunt Great Sabrecubs for their Lemontail drops, but that's going to take a good deal more effort on your part than this trick. Spiked Hares will spawn and die, dropping Meat and Cotton (another prized commodity for most people.) You'll routinely need to collect the droppings but its a nearly automatic method to collect a lot of dog/cat food, with the bonus of Cotton as well. If you run out of fish or meat, take 300 or so spike traps to Moonbrooke, find a spot the Spiked Hares spawn at (the area between the Hair Force's ship and cave to Rendarak is a popular spot), put down your traps, step back a bit, and let the game run for a while. Residents of the Elder job role, as well as Gillian and Finn, will fish for you if given a pool of water in their living area (and a chest obviously.) worstcase11 mentioned this but left off Gillian and Finn. I'm not sure if they'll eat fish that are inedible to humans or not (haven't tried this) So something like Red Snapper should advance them more than something like Lemontail. Higher quality fish or meat will make happiness grow faster. They can eventually grow to 100% on their own but at no more than 1% per game day.

(Since meat is much more plentiful than fish, dogs are quite a bit easier to breed.) It will take a couple hundred of either if you want to move things along at the fastest rate.

Dragon quest builders 2 pet food