Objective time is irrelevant, only subjective time matters
All your navigation calculations can be done on subjective time. All your shift patterns are going to work on subjective time.
You're far better off building a really good shipboard clock and doing good calculations relative to that, than attempting to automatically convert from some sort of external time prompt. Once you're out of contact with others, the only time that matters is ship time. So for these purposes, we're not going to correct for relativistic effects at all.
So what options are there?
For a traditional no computers environment, we need a good old fashioned mechanical clock. We can cheat and put an electric motor in the back so you don't need to worry about winding it, but we're better off rigging the motor against some nice constant force springs. That way rather than powering the clock itself the motor powers the winding mechanism. Of course this isn't going to be the most accurate option, falling at around 99.9977%
For a more accurate modern approach you want quartz. Something like the Casio F91W, the terrorists favourite bomb trigger due to low cost and legendary accuracy (99.9998%). Again it's a tried and tested system, the accuracy and time loss is a known quantity that can be calculated against. (For a premium system you can get up to 99.9999% accuracy, but where's the fun in that?)
However the trouble with both quartz digital watches or something like an atomic clock is the computer memory, a digital clock is fundamentally a single function computer. And in either case to know when to trigger or end your burns you need to be able to program a time into it to either count up or down to.
However it's possible to build an analogue clock with a quartz timekeeper* giving you a mechanical memory and mechanical output triggers. This should create your "ideal" shipboard clock without upsetting any of the computer zealots. The key difference being the amount of margin for error you're required to put into your navigation calculations at every stage.
Why doesn't absolute time matter?
In the simplest case, because you're going from "here" to "there". When you get "there" you can pick up your new vectors from the local space traffic control. The time where you came from is irrelevant, only the relative position of where you're going next. Your navigation clock doesn't even need to be a clock, it only needs to be a timer. The only times that matter for navigation are the time to next burn and the burn duration.
But we're an exploration vessel, we don't go to ports!
Now it helps if you have a separate clock for your mission time, but it's still not strictly required. You know where you are, you know the location of this place relative to the other places on your route and the calculations for their relative movements. It can all be calculated, it's going to get harder for every extra step you take as the margins for error increase, but it can be done.
This can be made easier with something that will make the over 70s happy. Lookup tables. When you're at point A and visible object B is on vector AB, visible object C is on vector AC, then you can look up your next journey AD, by knowing the difference between AB and AC.
* I won't quote accuracy on these clocks for now as they tend to be at the very cheap end of the market. I'm sure they could be made accurate but there's no call for it.