If you read the entire text of Hunter's Quarry and of Battlefield Experience, it becomes pretty clear that Battlefield Experience lets you have multiple designated quarries at once, not designate multiple quarries with a single action.
Except that Battlefield Experiences says
designate more than one not
have more than one designated. You even quote that in your own post. The importance of designate, the verb, being used instead of designated, the adjective, is huge. Designate is the act of choosing the quarry, not the status of something being chosen.
The only part of the text of Hunter's Quarry that Battlefield Experience overrides is that final, one-[sentence paragraph. It doesn't change anything about the minor action used to designate a quarry, nor does it provide any override or exception to the requirement that you can only designate the closest enemy (remember, even warlocks, who can curse multiple enemies, don't have a clause that says "if the nearest enemy is already cursed by you, you may curse the next-closest enemy).
The problem with this is that, even with your (collective, as many agree with your interpretation) you must change more than the last one-sentence paragraph. You must at the very least change the paragraph above that. Specifically the one that reads (bolded lines mine):
The hunter’s quarry effect remains active until the end of the encounter, until the quarry is defeated, or until you designate a different target as your quarry.
Indeed, if Battlefield Experience does mean what you are saying, they overwrote the wrong sentence. They should have overwrote that second to last paragraph of Hunter's Quarry, making Battlefield Experience say something along the lines of:
"Your Hunter's Quarry effect does not end when you designate a different target as your quarry. You may have up to your Wisdom modifier in creatures designated. If you are at your limit, you may end the quarry effect on one creature to designate a new creature. In addition, any attack made against a quarry receives a +1 bonus to attack rolls."
That's not to say I think you are definitely wrong, I'm just saying that if you are reading it correctly they should have worded it better (and I admit that even if I am reading it correctly, they still need to word it better). And really, I can see both sides of the discussion. I just think that the next time WotC releases errata, I hope that the clear up the wording one way or the other. (Note, I am not trying to bash WotC, here. I fully accept that a rulebook of this size is bound to have a few rules that are a bit ambiguous. I just hope that they clear this one up).
However, I still think that since they used "can designate more than one creature as your quarry at a time" instead of "can have more than one creature designated as your quarry at a time" is either poor wording on their part, or it means that you can designate multiple creatures at the same time (in other words, with the same action).
Of course, until WotC clarifies this with errata, the best answer is "whichever reading your DM agrees with" (unless you are the DM).