Relativity is the least of your concerns when different planets have different length "hours"

Nothing in the solar system has a relative speed different enough to earth to make common timestamps different - the difference is well below leap second impacts.

As for simultaneity we already have to deal with that at the millisecond level. An event logged at 16:00:01.000 as far as LA is concerned maybe logged at 16:00:01.080 in London, or maybe at 16:00:00.920

Space travel will be relatively easy since requires computers. Everyone will keep track of their local clock in seconds. Then they will calculate clock for places they care about, eg Earth-time, since those are pretty predictable. Relativistic corrections will be pretty minor.

My guess is that planets will keep local day-year. Mars is close enough to Earth that throwing in extra 37 min might make sense. OTOH, most planets and moon might use Earth calendar because local day cycle is too far off human internal clocks.

They will calculate time for other places, but will be less important than time zones since every communication will have light speed lag. No video calls, but lots of email.