Living life as a perma or huge bounty outlaw has consequences, so thats good. However, the outlaw refuge seems to have been setup to provide an alternate form of access to game services - e.g. bank, guild trader, fence, merchant.
Therefore, if the logic is that you can't really obtain town services, but an alternate form has been provided in the outlaw refuge for banking, selling to merchant, etc - why is there no riding trainer?
Because the one and only thing you can not do while having a bounty, is train riding skills.
Therefore unless you max your skills, there is no such thing as perma outlaw or even not paying your bounty daily so you can train your riding skill.
I would request Zos to add a riding trainer to the refuge. Either be consistent in that you don't have access to major game services while an outlaw, or you do. Having some, but not others, seems inconsistent.
**Note - you also can't list items on guild trader from the refuge unless you guild is the one in the refuge, but at least there's the workaround of putting items in bank, then having another char list them. For riding skills, there is no work around.