It sounds like you want a Mod to track what dungeons/trials that you have done this week (Or since you last reset the Mod). I think you would get better response from a Mod creator than you would from the Dev's for this QoL feature.
You can recur to a classic, old and commonly forgotten method of noting things on a paper with a pen. I still use it, since I am from the time when we used to drew the maps for Phantasy Star 1(1987) on Master System.
First, there is an addon for tracking when you last did a trial on a specific character. Not in game so cannot give you the name ATM. Ofc, only for PC/Mac users.
Second, the weekly should reset on the same day instead of last time you got the coffer. Zos already has the date and time established since it would/should be when the leaderboards start for the week.
easier coding, just change the reset to midnight 7:00 UTC every monday, dunno why they bother to write the cool down code track everyone separately to begin with.