We throw it many times (say, a billion) and obtain an average value of 3.5. Using that information alone

So my prior state of knowledge about the die is entirely characterized by N=10^9 and m=3.5, with no knowledge of the shape of the distribution? It's not obvious to me how you're supposed to turn that, plus your background knowledge about what sort of object a die is, into a prior distribution; even one that maximizes entropy. The linked article mentions a "constraint rule" which seems to be an additional thing.

This sort of thing is rather thoroughly covered by Jaynes in PT:TLOS as I recall, and could make a good exercise for the Book Club when we come to the relevant chapters. In particular section 10.3 "How to cheat at coin and die tossing" contains the following caveat:

The results of tossing a die many times do not tell us any definite number char- acteristic only of the die. They tell us also something about how the die was tossed. If you toss ‘loaded’ dice in different ways, you can easily alter the relative frequencies of the faces. With only slightly more difficulty, you can still do this if your dice are perfectly ‘honest’.

And later:

The problems in which intuition compels us most strongly to a uniform probability assignment are not the ones in which we merely apply a principle of ‘equal distribution of ignorance’. Thus, to explain the assignment of equal probabilities to heads and tails on the grounds that we ‘saw no reason why either face should be more likely than the other’, fails utterly to do justice to the reasoning involved. The point is that we have not merely ‘equal ignorance’. We also have positive knowledge of the symmetry of the problem; and introspection will show that when this positive knowledge is lacking, so also is our intuitive compulsion toward a uniform distribution.