No, that's not the problem. You don't know how much money is in either envelope, but it needn't have been selected randomly. The only randomness is whether you the envelope you choose has the greater or lesser value, which is just a uniform distribution on two points.

That being said,

>It is impossible to choose a value from an infinite series (the list of possible monies), with every value having an equal opportunity of being chosen.

this is also incorrect. You're right to say that you can't have a uniform distribution over the possible values of money, but that's not just because it's infinite. You can have a uniform distribution over [0,1] for example, which is larger than the set of possible money values.