You didn't outline the reason why you don't want to use the time-tested and well-validated approach -- and you even identified it as efficient, which usually means it has a good cost vs. quality balance. And based on your questions, it might be worth revisiting why the classic experimental method runs several hundred cases.
In an unsteady flow, things are often quite chaotic due to turbulence. Small perturbations in conditions can lead to pretty significant changes in the flow. Many of these changes are seemingly random and things like intermittancy or bursting can occur at very irregular intervals.
By making a measurement many, many times and averaging them (ie. ensemble averaging), you eliminate most of those perturbations in your initial and boundary conditions. That means that whatever small changes that lead to chaotic flows will usually all average out, and you are left with a "true" representation of the flow based on the expected values of the initial and boundary conditions.
By trying to bypass that, you are introducing a possible source of uncertainty into your results. Let's take your example of measuring just once. It is possible that when you take your measurement, an intermittent vortex is shed somewhere upstream of where you are measuring and that passes through your measurement point. You may have gotten unlucky and that is the only time that vortex has ever or will ever appear, but you happened to measure it. So now your only data signal says a vortex will always be there. But, if you measured 200 times, your data would say that the vortex is unlikely and the results could be different.
Similarly, you are interested in the mean velocity from a time series. That means if you only take a single measurement, it had better be a long (in time) one. How long? Well, it depends on all the possible frequencies of fluctuations in your flow. If you are measuring a river, you might need to measure for a year to account for all of the seasonal variations. If you are measuring a supersonic jet in crossflow, you might only need to measure for a millisecond. You need to make sure your time series is long enough to resolve every possible frequency you want.
So be careful... you might just end up making a lot more work for yourself by trying to bypass the traditional techniques.
To answer your specific question... the mean of a signal is always the component of the FFT with zero frequency. No frequency implies mean. But again, be very careful that your data isn't taken when there is a slow rise or slow decrease of a low frequency trend. You need a long (theoretically infinite, but always truncated in practice) dataset in order to find the true mean.