In that case the organics would be from the water, anything left in the filter and possibly even the vinegar you used to clean the tank.
I've just looked at the Edge instructions and see that the filter contains three types of media - a bag of ceramic noodles (biomax), a foam pad and a 'carbon clean and clear' cartridge.
I would ignore the clean and clear cartridge as you don't need carbon full time.
Then once you are ready to move the fish, squeeze as much of the Edge's sponge and biomax into the bigger tank's filter as possible. You can cut the sponge up to make it fit, and leave the biomax in its mesh bag. If there's any space left, fill that with the media that's supposed to go in the bigger tank's filter
Then move the fish.
I would also move any decor over even if you want to have different decor eventually. The bacteria you want to preserve live in the filter media and on all the surfaces in the tank, including the decor. This is why I suggest moving any decor over in the short term. You will lose the bacetria on the Edge's glass and other surfaces you can't move so every single bacteria you can move is worth it. Once you know the transfer has gone smoothly, you can replace any unwanted decor a bit at a time.
Feed the fish lightly for a few days before and after the move and nothing at all on move day. And monitor ammonia and nitrite for several days after the move. If you do see either of them, water changes will reduce them, and the mini spike shouldn't last long.
However....
I notice that the Edge's filter is a HOB (hang on back). Is there any way it could fit on the new tank?