Okay, it's a pretty raw tangle of thought experiment, but here we go: You can't. There is no such thing as multiple infinities. Imagine if there were, all such multiple infinities would simply exist within a "larger" infinity that is ultimately singular. That ultimately singular one is the only true infinity, as any smaller ones are "not as infinite" in comparison to it, and cannot rightfully be considered actually infinite, although they could be considered as approaching infinity.
Consider having a cake, with a book beside the cake that contains 144 ways to cut the cake. Now cut the cake according to one of the descriptions. Once the cake is cut, one way is implemented, and 143 other ways are not implemented. The others may exist "in theory" within the book, but "in practice" only one can ever exist.
The only way around this is to assume that time can be manipulated, and after one way of cutting the cake is implemented, time is somehow reversed, or bent, and thus the 2nd way can be implemented, etc. But if time can be manipulated in order to define multiple infinities, then what is the point of making a definition?
Your adherence to the idea of multiple infinities is admirable, but unnecessary. Rather than chase your tail trying to encompass the idea of multiple infinities which are so truly "infinite" they have no reference to each other (except in the thought experiment which assumes their existence, the existence of which itself creates a set that contains them all), simply define infinity as the top-most infinity and start from there. It contains e v e r y t h i n g, not just Cantor's many numerical infinities, which drift off toward potentiality and eventually escape the bounds of actuality, but also all of Nature, the entire Universe, and all within. Everything. That is the only true Infinity, and within that infinity is the idea of multiple infinities (but not the actuality of multiple infinities.)
I believe the idea of multiple independent infinities is an expression of ego, an attempt to place oneself in a place which is properly the throne of God, a being who encompasses all, and can operate freely upon all. I believe this because all operations upon infinity necessarily assume there are multiple infinities: or at the very least, the one being operated upon, and the one doing the operation, since you cannot, for example, infinitely divide an infinity from within itself.
The infinity doing the operation must contain everything within the "smaller" one, and more. The only one who can contain all in this manner is God himself. He alone can define whether there are multiple infinities, and for his opinion on the subject, we have the ensample of Jesus Christ, who died "once" for all. If there are multiple infinities, then Jesus Christ would be dying multiple times (or else the infinities are inferior to this one).
Supposing the case of a recursive infinity, where the "top" and the "bottom" of infinity (the Singularity and the sub-Planck-length? particles) are one and the same, with the difference between the two being the way each view is perceived (a perceptual difference rather than an actual difference), you have 3 infinities: The top, the recursively linked bottom, and the perceptual one that allows for multiple simultaneous views of the same thing.
Bother, this hurts the brain to try and think through this stuff, as there is a switch between binary and ternary logic which I keep implying but haven't been explicit about, and it gets confusing if you don't do this kind of thinking very methodically. Well, the headline captured the point I wanted to make anyway. G'day.