Im really not sure I want multiple clocks on board, it sounds like a complete convoluted mess. If every player was located not only in a different space but time also, how are you going to simulate an economy when technically I can arrive at port after you but buy the stock you just bought, before you got there. My head hurts just thinking about it.
It's simplicity itself - you have your ship's clock which shows your personal time, and also a second clock which shows the time being used at a target location.
Ships' clocks could remain in GMT format, (24/7/365), with personal +/- deviations between individuals depending on their own accrued dilations.
Local timezones would remain specific to themselves, and adjusted to players' own deltas upon arrival.
So there's no contradictions - you'd use local timezones if you wanted to meet another player or NPC, and obviously any such appointments would need to be in both parties' futures. So if your ship clock says it's 6am May 1st 3302, and mine reads 16:30pm July 22nd 3301, we can still agree to meet at Abraham Lincoln station at 12pm November 6th 3302, since it's in both of our futures. The fact that you'd have aged a year older than me would be irrelevant.
If OTOH we mistakenly chose a date a year earlier, 6th November 3301, then i could get there but you wouldn't be able to, since it's in your past. Likewise, if we chose a date in both our pasts then we'd both be late.
So you'd have a time-dependent instancing system. It DOES add a novel layer of complexity compared to our usual timekeeping schemes. But from our computer's perspective they'd just be implementing very simple rules, that we'd quickly get used to.
If an interstellar civilisation ever arises, we'll have to deal with these practicalities anyway - it's hardly going to render business impossible. Compared to the hurdles of insterstellar propulsion it's small potatos.
More to the point, for a game such as this it adds a level of nuance, subtlety and immersion that's entirely befitting, manageable and which broadens the depth and scope of gameplay.
A blanket universal time for all locales, with no individuality, is just cartoonish. It's childishly naive. A minimum of two clocks could work well, without invoking any anachronisms.