Your desire for such an algorithm cannot be satisfied. As randomtoad said, it will require more moves than the best odd parity algorithms that exist already. Your best bet is to just do double layer turns with one of the pure algorithms (like everyone already does).
I really appreciate this expose, and am in agreement with your take on this parity problem, but we appear to have a different understanding as to what this "algorithm" is supposed to be able to do.
Maybe to clear this up, I should first state what this New dedge flip algorithm is NOT supposed to do. That way we can all be working together here.
1) This algorithm does NOT have to restore the flipped dedge to its original location. I know the edge pieces need to be swapped and then paired back to reform the dedge, but the permutation of this piece can be any dedge position on LL or the yet to be solved dedge for F2L pair. So any of these 5 possible spots to return the dedge to is fine. This allows for more flexibility to find better base edge exchange algorithms that can be used to make the actual dedge flip.
2) F2L is solved except for the last slot pair, and LL is open. This algorithm then does NOT have to worry about making changes (psuedo 3x3x3) to any of these 10 locations. This should again make it easier to do the flip algorithm since changes are not only allowed here, but also do not have to be restored back again later. Another aspect of this restriction removal is that there are 5 corner pieces that are happy with any orientation, permutation, or parity situation. This is a HUGE advantage in terms of finding good finger friendly algs, as now the single slice moves (i.e r,r',l,l',u,u') can sometimes be coupled with their respective face turns making for example, changing [r u r' u'] into [(Rr) (Uu) (Rr)' (Uu)'] without messing up anything important.
3) This algorithm does NOT even have to flip just one dedge, and in fact can flip any odd number of dedges (1,3,5). If it turns out that there is a shorter pure edge exchange that works on more pieces (not likely, but still possible) then the parity issue would still be solved. In other words, it makes no difference how many we "flip", just as long as it is an odd number of dedges that get the treatment.
4) This algorithm must NOT unpair any of the dedges, as this would obviously be counterproductive. (Stated for the record).
5) This algorithm must NOT mess up the centers. (again for the record).
Are we seeing eye to eye on this now?
The ideal algorithm, which you are interested in, can be derived from an pure “edge flip” algorithm in which its ending moves are only outer-layer turns, there by making it possible to omit those last moves so that you are not doing worthless 3X3X3 restoration (as long as those moves restore only the last layer and maybe one F2L slot, as you have requested).
You probably see now that it is possible to create a better "unpure" edge flip algorithm that would satisfy the much looser conditions of this problem.
There is no "dedge flip" algorithm that exists (which is shorter in length than the pure edge flip algorithms) that satisfies this constraint, nor will there ever be.
This "proof" has some solid logic to it, BUT there is NO constraint for using only "pure" edge flips, or on having to restore the dedge to its original location. I believe that both better single edge swap algs and better dedge parity flips based on them will be found.
Proof:
There are two main reasons for this:
[1] "Edge flip" algorithms', or even just every pure odd parity algorithm’s, main goal is to preserve the centers: if they don’t preserve the centers, then you might as well use the Cage Method (which as you know is not going to give you a world record: commutators are an inefficient way to solve the centers).
[2] The winged edges need to be separated in the beginning of the algorithm and subsequently at the end of the algorithm (which is slice turns, not outer-layer turns).
Unlike all other forms of odd parity restricted to the LL, the "one edge flip" is the case which requires the most moves (or at least, no other form of odd parity will actually require more than the least moves that the one-edge flip can be done in) because the two winged edges are initially in the same composite edge.
The "r2 B2" and "B2 r2" that you see at the beginning and end of common algorithms is to bring the initially front-right winged edge to the back-left, so that the front-left and back-left are directly across from each other (or vice versa, depending on which mirror you do). (Note: there are other ways to move two winged edges besides r2 B2/l2 B2, which are originally in the same composite edge, opposite to one another, in 4 block quarter turn moves, but this is the only way to get them across from each other in 2 block half turn moves).
For example, the best opposite winged edge swap from my understanding (or its mirror) is:
l' U2 l' U2 F2 l' F2 r U2 r' U2 l2
Now, to make this a pure edge flip, it is going to cost us some more moves. We start off with adding "l2 B2" to the beginning and "B2 l2" to the end.
l2 B2
l' U2 l' U2 F2 l' F2 r U2 r' U2 l2
B2 l2
But, we have a problem: the top and bottom centers are discolored slightly.
To fix this problem, all you need to do is add 1 half turn slice in the beginning and the end:
l2 B2 l2
l' U2 l' U2 F2 l' F2 r U2 r' U2 l2
l2 B2 l2
As you probably can see, there are three block quarter turn moves which disappear due to move cancellations, and we are left with a 25 block quarter turn alg/15 block half turn alg:
l2 B2 l U2 l' U2 F2 l' F2 r U2 r' U2 B2 l2
Just as the "base" for the alg above was l' U2 l' U2 F2 l' F2 r U2 r' U2 l2, if you try, you will find that the base of r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2 is:
r2 B2 r2
r2 U2 l U2 r' U2 r U2 F2 r F2 l' r2
r2 B2 r2
Very nice expo on the logic used to create these algs. I appreciated that!
As Lucas Garron mentioned, maybe what you are really hungry for is a pure edge flip algorithm that is the least number of moves. I am assuming he meant block quarter turn moves, because in block half turn moves, we already have the optimal algorithm (15).
For speedcubing purposes, these briefest pure “edge flip” algorithms (in block quarter turn moves) are most likely not going to be necessarily faster to perform than the current algorithms (I perform a few of mine around the same amount of time), even though they might be 1-2 block quarter turn moves less.
What I want is the dirtiest, nastiest, pure, or unpure, single "edge flip" you can come up with. As long as it satisfies the conditions above - I will be VERY happy. It will not be a problem at all for me to invent the cube rotations, inversions, conjugations, and cancellations to optimize from there.
reThinker
