Efficient revealed preference check of whether two preferences over lotteries are the same?
Perhaps I'm way off base on this, but given the relatively strong requirements imposed in the question, I'm tempted to say that we can never truly be able to state with certainty that some individual's utility function $u_i$ is absolutely a linear transformation of some other function $u^{*}$ considered ex ante. While depending on the specific $u_i$, $u^{*}$ and initial questions asked, it could be possible to dismiss the two as not satisfying the transformation requirement, but I don't think one could ever actually confirm it.

The underlying reason for this thinking is the inclusion of vNM utilities and lotteries over outcomes. Since the former assumption imposes cardinality on our utility framework, there is always the possibility of discrepancies between $u_i$ and $u^{*}$. Even considering just one good (money, for example) any sequence of observed choices made by some individual following a utility function $u_i$ might only differ from how an individual following $u^{*}$ behaves in the "gaps." No matter how fine the probabilities and dollar amounts get, we can always come up with some alternative $u'$ which replicates the decisions up till that point. (We can always consider, for example, some $u^{'}$ that's a piecewise on some interval where $u^{*}$ is smooth. 

Of course, as our ability to observe smaller and smaller intervals increases from repeated choice observational data, the possible deviations between $u^{*}$ and some alternative $u'$ trends towards zero. But again, finding out that all observations thusfar suggest the individual's utility $u_i$ is a linear transformation of $u^{*}$ doesn't guarantee that there isn't some interval of the function where they're different.

It should also be pointed out that this really only considers extreme cases- for all experimental purposes we only need a function that approximates an individual's utility function. But I do think it's tougher to truly prove one is a linear transformation of the particular $u^{*}$ in mind.