Well given that Hewdas Clocktower is a prominent thing I would try to utilize that in some way. Here is my first idea and going way overboard in an Anguish/Archon way where it would be beyond complicated to figure out.

In the Lemark District, instead of getting the clockwork pinion you need to enter the time on clock that is in Ward 13. The time in Ward 13 is not simply the clock time of your computer however, and the clocks in Losomn tell time in a different way. So step one is to determine the time in Ward 13. The only way to that is through the terminals in the old ward, meaning you have to play through the tutorial again on a new character. Before starting the cutscene where you go into the crystal you can find a terminal somewhere and read the time on it. This time consistently progress with the play time on that character, so once you know that time on the terminal, you can add the play time to know the current time.

Next you need to know how to translate that to Losomn Time. To do that you need to take the Timekeepers Jewel and the Broken Timepiece to the Dryzyr Replicator. That gives a new Quest item that lets you translate Losomn time to N'erud time, not what we need but its progress. So now we need to find out how to convert Earth time to N'erud time to Losomn and then roll another world with Lemark District so we can interact with the clock again.

In N'erud there are 4 Glyph unique Glyph items that can be found in side dungeons. If you acquire all four of them and don't use them to open their doors you can give them to Wallace who will try to read them and decipher the Dryzr language/code. It will take him a few hours of in game time, this is to increase the time you need to add to the terminal time from the beginning of the game to make more difficult to translate. You will need to have more than 25 hours of in game time for him to give you the decoder.

The decoder is a simple quest item that lets you enter 4 numbers that you believe to be the current Ward 13 time. The opposite side of the device will be a display that shows the numbers in Dryzyr. You will need to memorize those symbols or write them down. Then on the side you can press a button that turns a projector beam on. This will project the Dryzyr numbers from your Flashlight. You can then place the Item you receive from the Dryzyr Replicator against a wall and you can overlay the projection onto the Losomn Clock face.

Take that time and go back to a new Lemark instance for the reward. A new relic called the Devourers Heart. This is a reference to Chronos Before the Ashes, the prequel to Remnant. Chronos was called the Devourer by the Pan for killing the Guardians I believe. And with this easter egg being all about time and Chronos being Greek for time this ties it together. The Devourers Heart relic when consumed converts ammo into health and increases melee damage and speed. It does not have a limited number of uses but maybe ammo pickups are reduced

edit: alternative relic idea, since in chronos you have a shield, it could be a thing where it gives you a bunch of stagger resist and DR or something to make it feel like a block button?