Heck yep, Gokudera can definitely ride with him! Hijikata's horse is a diva and doubtlessly demanded plenty of apples and carrots for the trip, but it's also big and badass-looking.

A cloak and/or an everwarm potion sounds like a good idea to bring along for sure! Hijikata will bring his own (standard-issue) winter cloak and some blankets too, just in case. He spent a night on the mountain with Crow in spring when they brought some of the dragonlets home, and he never wants to experience such cold and uncomfortable sleeping conditions again!