I know that there are checklist cards that can be used in lieu of pulling a double-faced card out of its sleeve every time it needs to be flipped. Would it also be legal to simply use another copy of the same card in an opaque sleeve, then trade it out for one in a clear sleeve when it gets played?
In general, yes. Nothing in the comprehensive rules or the Magic Tournament Rules disallows this explicitly; rather, as long as the card in the clear sleeve is kept separate from the deck and sideboard, it may be treated no differently from items such as tokens and counters (for example, as a "physical, visible, and reliable method to maintain ... game information" under M.T.R. 1.10 in this scenario; see also C.R. 100.2).
Under C.R. 711.3, "[p]layers must ensure that double-faced cards in hidden zones are indistinguishable from other cards in the same zone". On the other hand, C.R. 713 and M.T.R. 3.5 apply only if a player uses substitute cards (including what were formerly known as checklist cards), and not if a player represents transformed permanents by other means. And otherwise, there are no special requirements in the comprehensive rules for representing transformed permanents in public zones, such as on the battlefield (C.R. 400.2).
See also this thread, which involves putting the alternate card on top of the original, rather than "trading it out", while it's transformed (not: flipped).
EDIT: Edited after comment 4 was posted.
EDIT (Sep. 24, 2020): Edited to conform to rule update with Zendikar Rising.
It might cause issues for having more than the listed cards in your deckbox. In a competitive setting this could get you in trouble.
Using Checklist cards already does this. If you have 4 checklist cards in your deck, you need to have 4 real cards available. Not using checklists doesn't prevent you from using additional copies of the real cards; they just need to be kept separate from everything just like they are when using checklist cards.
As long as the additional copies are in a different colored sleeve than the deck and sideboard so they cannot be mistaken as being part of the deck, it is fine to have 4 copies of a card in the deck and 4 copies not in the deck to be used for just their back face.