In my current playthrough, I did Lon Lon Stick and used RBA with Pocket Egg, Odd Mushroom and Odd Poultice. I'm not sure what exactly I did afterwards, but it was nothing significant other than running around, save & quit, Light Arrow cutscene, trade for Poacher's Saw and travel to the Ice Cavern, but in the first room of Ice Cavern I suddenly noticed I could use the sword again, even though I didn't get Giant's Knife, so I guess there has to be some way to get the sword back without reequipping it? I didn't play any other files during that time, either.
Could be because you did something after the light arrow cutscene had been triggered. On console the cutscene used to trigger BA, but that isn't the case on 3ds.
Maybe you died and/or saved sometime after the light arrow cutscene.
I never really tried to do stuff after the light arrow cutscene to see if I could get the sword back.
Currently we do not know any way to re-equip the sword on a file that you've
never equipped a sword on. Like a Lon Lon stick file for example.
Here is an idea I had a long time ago for how the route could be changed:
GET 10 SKULLTULAS AS CHILD
TSC escape
Cucco bottle
KZ skip
Farore's Wind after jabu
odd mushroom after jabu
Wallet
200 Rupee Chest
Giant's Knife
Mushroom
Set Warp point in Meadow (you do this because you have to wait for timer to run out anyway.)
Get Swordless (Archery method)
RBA Odd mushroom
Warp back
Note this was ever timed. It was just speculations. I think it could have some potential. I have no idea if it will be any faster, but even if it isn't, it should be quite similar, and it will remove the stupid Lon Lon Stick glitch. (yay for kokiri sword).
I took some estimate times (not actually timed, just made up in my head), and it feels like it should be faster.
Someone should try it out, or time each differences in the run and see how it ends up. You do not have to wait for egg to crack, no King zora CS, you do not have to buy all the nuts at the start. You get kokiri sword which means that you do not have to run around and collect dekusticks.
On the other hand, you have to get odd mushroom, and other things.
So it should even up at the end, hopefully with the new route ahead.