This one's easy. Chuck the 4l vessel and just brew 4 1l batches.
Sometimes the simplest solution is the best. I personally would rather try to figure out why the 4L isn't working right; the thrill of the hunt.
But it's hard to deny this logic. If 1L volumes produce the product you like, in the time frame you require, it makes sense.
The size and shape of the container you are using can influence the end result.
There is a reason why commercial fermentation tanks are specific height to width ratios, and why conical cones are at the angles they are.
If your 1L vessel is tall and narrow, and the 4L is short and wide, this can influence the way yeasts do their thing. I'm not a high enough level brewer to explain the how and why, but it comes up from time to time from people who know what they're talking about.
Look at the ratios for the two containers, and try to find a new large container that mimics the 1L closer.
If the 1L is 1" by 4" (this is too small for 1L but it's an example for math's sake), then the ratio is 1:4. Find a larger container that is 2" by 8", or 3" by 12". Typically it seems the wider shallower containers make a more sour KT, but you wouldn't want to use a cookie sheet. I doubt that would turn out very well, not to mention it would be hard to cover lol
A suggestion you might want to try. If you put 50ml of vodka in 3L of fermenting KT, it sours really quickly. I've got a document around here some place of and experiment I did with vodka in primary. It doesn't take much, and the bacteria get started on it right away. The problem you may have is, if your yeast aren't performing well, you'll end up with cloying vinegar.
Fix the yeast problem first. Get an empty 20oz soda bottle and fill it half way with finished KT. Dissolve a tsp of sugar in a half cup of warm water, and let it sit out until it cools to at about the same temp as the KT. Then add it to the bottle. Make sure there is at least 1 inch of headspace in the bottle. Then squeeze the bottle with the lid off until the liquid is up to the top. Put the cap on it tight. Every couple days, check on the bottle. The yeast should process the sugar, and release CO2 which you'll capture. If the bottle doesn't fill back out, you have under performing yeast. Or none at all.
Are you using tap water? If so, you need to dechlorinate it before using. Chlorine is added to kill off things in the water. You'd need to boil it, or let it sit out uncovered for an amount of time. You can also treat the water with products designed to remove chlorine, like Campden tablets.
The temperature you ferment at greatly influences the yeasts and bacteria as well.