This one's an absolute gem. Do have a go before reading on.
The solution is long and involved, and I don't have a program handy for fancy animations, so I'll only give the general outlines, with more detail on the critical bits.
First things first, we have only one chance of guiding the Pink Lady Stickotaur of Universe A (henceforth called the "A-taur") to the trap at r2c1.
So we start with LL.
Then, since we're not counting moves, the A-verse is free for us to roam, and we eat all the apples in Universe A, except for the first row and the square the A-taur is in.
There's absolutely no way we can eat the apple at the A-taur's trap and then reach the portal afterwards, so it'll be necessary to end our journey in universe A. Keeping that in mind, it's time to enter universe B at this point.
And boy, are we in a hurry. As we materialise in universe B, the B-taur is instantly freed from the promising nook it started in. Time to run. UUUR barely saves us, and traps the B-taur at r2c2.
This lets us clear out universe B except for column 1 and the B-taur trap. Those look like a real problem, since there's no way to lure the B-taur, well, basically anywhere without getting caught. But maybe we can use the teleports for that later..
In any case, we'll need to head to universe C at this point. Since the lower teleport is closely guarded, we'll want to poke our smiling head out at the top teleport first, and immediately teleport back to the B-verse.
Now we can re-enter universe C via the lower teleport, and move right to barely survive.
While the C-taur is trapped on the top row, we can clean up the C-verse, except for row 1 and column 1. Those will require some superb precision later, since after we go to r4c2 and move the C-taur into another prison by LRRL, we can of course easily clean up column 1, but getting the apples in row 1 would require stepping onto r1c3, which is taboo: it would release the C-taur.
Here's what the multiverse looks like at this point, after the "easy parts":
Since there's nothing else we can do, we'll return to universe B via the lower teleport at this point, and then do
The absolutely brilliant part
Eating the top row apples in universe C would release the C-taur, rendering the universe C shortcut inaccessible for us. But there's absolutely no way we can ever grab the lower left corner apple of Universe B without teleport shenanigans. After racking our wormy brain for quite a while, we finally figure it out: we can to move the C-taur to a better prison! We take the upper teleport back to universe C and PASS. (Yes, stalling for a round is allowed by the rules. If it weren't, we could teleport back-and-forth once for the same effect.)
Now the C-taur is nicely tucked out of the way at r2c4, and we are finally ready to tackle the rest of Universe B.
Universe B, top part
Now we can take the lower teleport back to the B-verse and release the B-taur with LR, and mysteriously vanish into the teleport.
While the B-taur is standing mystified at r4c2, we sneak around to the other teleport through universe C, finally emptying universe C of apples on the way, and when we reappear in universe B, a simple move right will trap the B-taur in the spot it started on. We clean up the apples left in the top three rows or universe B, and then saluting the brilliance of the puzzle creator, we head back to the upper teleport to universe C. The B-taur will be back at its starting spot when we leave.
Universe B, final apple
With the C-taur still neatly trapped at r2c4 (because we managed to figure out the brilliant part), we take the lower teleport back to universe B, and just barely manage to eat the final B-verse apple with LER. We then teleport back to the C-verse, and return through the other teleport. A step right will again trap the B-taur, and we can finally return to the A-verse.
After the intense excitement of the other universes, we take a deep breath, go to r2c4, release the A-taur with UDL, bring it to its new jail by DDDLL, after which we can eat the rest of the apples at our leisure.
What a journey!