Your character data on FFXIAH updates only three ways assuming you have "Public All" checked on your Linkshell Community Profile.
1. Automatically data is pulled and updated to your profile every 36-48 hours based on what you last left your character from that night before.
2. You can do a force update while logged in your FFXIAH account by clicking on "Que for Update" this will pull the data last updated from the Linkshell Community Profile to your FFXIAH profile any new data will only take a few seconds to apply.
3. If you have a properly linked Guildwork account, you login to Guildwork via in-game (that generally automatically flushes you) or you can type the following command...
/gw flush
This will update any and all job levels, merits, titles, and any newly received gear.
Wait about a minute, refresh your FFXIAH profile, and you should clearly see the changes.
All of this only works with Linkshell Community Profile being set to "Public All" and having a Guildwork Account properly linked to your Linkshell Community Profile.
Although Linkshell Community is still under maintenance all of its basic rudimentary functions are still active far as data dumps go.
I hope this was at all helpful, Good Luck!