I'm not sure that the use of the phrase "as it enters the battlefield" implies that it's on the battlefield when the choice is made. Actually, I've always taken it to mean something like: the choice is never relevant before it's on the battlefield, but always relevant after.
Certainly, there's no transitional state in which the choice would be made, but that doesn't preclude players from making a decision at the same time as it moves. The comprehensive rules use the word "as" (in the sense and context of timing) infrequently, but when they do, it tends to imply a transition.
For example, consider the following sentence from the comp rules: A card or token becomes a permanent as it enters the battlefield and it stops being a permanent as it’s moved to another zone by an effect or rule.
Certainly, in this sentence,
"as" cannot mean the thing happens before -- a card is never a permanent before it enters the battlefield. And it cannot mean after, either. A card is never on the battlefield and not a permanent. The word literally implies that the card becomes a permanent during the transition.
I believe that "as," in the context of Phyrexian Metamorph
means the same thing. You don't make the choice, then put it on the battlefield, nor do you put it on the battlefield and then make the choice. They happen together, and to imply that one happens before the other makes about as much sense as something in your hand being a permanent.