So pretty simple then, you don't need an RTC, just roll your own,
I would organize a 1/nth-second timer and two counters that count modulo 60/60/24. Then every 1/nth of a second update one counter (this is you fast counter) and every n interrupts update the other one (the real time counter).
So now we have two counters running counting H:M:S but one is n times faster than the other.
When someone interacts with the device set a flag (and possibly reset the fast counter with the value from the real time counter, depends on what you want to show)
Write a routine to display a "time", it gets called every nth of a second, it tests the flag and either displays the fast or slow counter.
That's pretty much it I think, plus of course you have to be able to set the time so that will need a few buttons, dip switches or something.