There are verbal suggestions everywhere on when one should use a geometric average or when an arithmetic average should be preferred, but I can't find any formal statistical treatment of this question. Is it possible to formally test which one of these averages should be used for a particular sample?
Such decisions are not normally made on the basis of testing, but on an understanding of the variables, the circumstances and the needs of the analysis.
For example, we would need to consider when it is more meaningful to us to use a geometric mean and when it is more meaningful to use an arithmetic mean.
If I am interested in a population mean, generally speaking I probably want to consider a sample arithmetic mean. However, if I have particular distributional assumptions, it's possible a geometric mean may (perhaps after some transformation, or in combination with other quantities) lead me to a better estimate of the population mean than the arithmetic mean (e.g. in a lognormal distribution, the geometric mean might be part of the calculation of a better estimate of the population mean than the arithmetic mean).
You seem to be suggesting that we can infer the best estimator from the data, as if we perform two stages of estimation, first of the distribution, and then from that choose a good estimator.
But in fact it's not at all clear that this is generally a good approach to estimation, except perhaps in certain classes of estimation problems, with particular structure (this is essentially a form of adaptive estimation). If you want to pursue that approach you'll likely need to narrow the scope of the problem and do some kind of study (either by looking at asymptotic properties or by use of simulation) of the properties of some proposed adaptive estimator.