I don't think there've been many court cases exploring the meaning of "[to] fix the Standard of Weights and Measures", and probably none as it regards time. And in the modern era the Commerce Power would probably be sufficient on its own. SCOTUS has suggested that under exiting precedent Congress would have the power to grant copyrights and patents under the Commerce Clause, and so the Copyright and Patent clauses today act more like restrictions on Congressional power.
But I don't see a problem relating time zones to measurement. Part of the authority to standardize measurement is the ability to dictate the manner and means of determining a quantitative value. Under the Weights and Measures clause I think Congress can regulate things like scales, including their precision and accuracy, at least in so far as they claim to provide a measurement of a Federally standardized unit. You might intuitively think the only reasonable end to such power is using it to improve and mandate ever greater precision and accuracy. But sometimes too much precision and accuracy is a bad thing--it can create transactional friction. Case in point, when 12PM noon varied between every town it become increasingly problematic as the speed of long-distance transportation improved, i.e. the rail roads. So the solution was to mandate worse accuracy.
Relatedly, there's a whole separate question of what time means. Most HN readers understand time in the scientific sense, and think of time in the sense of the SI second. But civil time used for general daily life has a slightly more nuanced meaning. That said, UTC/TAI time is very much like time zones in the sense of fudging accuracy. Modern clocks and gravimeter, even the kind regular people can buy for a few hundred or thousand dollars, are precise enough to be able to distinguish local time dilation. So the time passing in your living room is actually different from UTC/TAI. But think of how complex and, for the most part, useless it would be to try to "solve" that discrepancy by trying to integrate that reality in the general definition of civil time.
Also, AFAIU the authority to standardize measurement, and time specifically, operates more as a prohibition on states imposing their own mandates. See, generally, the Legal Tender Cases for the push and pull between various powers allocated between the federal government and the states.