First, I do not see why this is a Bayesian construction. Indeed, if you do mean that the density is known, it cannot possibly be a Bayesian construction. Consider the case where $f(x)=280{x^3}(1-x)^4,x\in[0,1]$,. There is no unknown parameter here. This is both your prior and your posterior as no amount of data will alter anything.
If this were a Bayesian construction then there would have to be some uncertain parameter, but there is no uncertain parameter. This is a beta distribution with $\alpha=5$ and $\beta=4$ The prior is forced to be $\Pr(\alpha=5;\beta=4)=1$.
The question is "what is x?" The only uncertainty is in the valuation. This is a Frequentist problem.
EDIT
In the case where it is drawn from an unknown distribution, you are facing two options, even if the distribution is known with certainty to the actors.
The first is to use Bayesian non-parametric methods, the second is to use Frequentist non-parametric methods. Depending on what I wanted to accomplish, I would choose one or the other.
The Bayesian method will be coherent and so you could place gambles on it. It will also likely be very difficult to implement. There cannot be a Bayesian solution that is free of its prior. Such a thing does not exist. It might be that it is uninformative, but it must exist. The alternative is to use Fisher's failed method of fiducial statistics. The Frequentist method will minimize the maximum loss you could experience from making a choice based on the data by using an incorrect inference. It will also allow you to control for power. It will usually be far simpler to implement.
Bayesian non-parametric methods are potentially infinite dimensional constructions and you would need to do a bit of reading on them. A simple approximation though would be to use the beta distribution because of its incredible flexibility, although you could use any high degree polynomial that stays above the axis since your bounding guarantees that a constant of integration exists. You would then perform model selection.
As long as you believe it is unimodal, the bounding on both sides guarantees that a mean exists. Even though your distribution is unknown, it is guaranteed to have moments. The t-test is probably inappropriate because of the bounding is so tight, but you could use the empirical quantiles to test significance. If you felt you needed the higher moments, the method of moments is always available.
Finally, in either case, you have kernel methods available to you.
You cannot avoid a prior using Bayesian methods, but the greatest advantage of Frequentist statistics is to be able to solve problems when you cannot form a prior.