Speaking as a potter, do not use the wire pad.
The pot you have is a very low-fire ceramic, and the glaze is most likely softer than steel, so a wire pad could scratch the inside of the pot irreparably. Note that this is not true of high-fire ceramic, which is harder than steel (as is glass).
There is no issue with soaking the pot, really. Water will not hurt it. It is possible, even likely, that the clay will soak up water through its bottom or tiny cracks in the glaze. In this case, you just need to make sure the pot is dry before exposing it to sudden heat, or it could crack. Dry it on a dishrack, or even in a low-temperature (less than 200F) oven. Also, don't use very much soap while soaking it, as the pot may soak up some of the soap and affect the flavor of the food.
Additionally, several companies make plastic or even copper scrub pads. I've found, in particular, that the red-and-yellow plastic scrub pads made by Tuffy are particularly effective for scrubbing delicate surfaces. I don't know if the copper scrub pads are pure copper, so I don't know if they will scratch your pot or not.
Finally, let me also recommend against putting your gyveche through the dishwasher. Not only will this expose the pot to almost certain chipping, but the porous earthenware may soak up bleach which can both ruin your food and cause the clay to break down and chip or crack.
By the way, that's a beautiful pot. I can see why you like using it.
The "solid vegetable oil" you're describing sounds like Crisco (shortening), which you can find in any US grocery. You could also try refined coconut or palm kernel oil, both of which are solid at room temperature.
Another possibility to note is that the butter may be fine, but maybe the chickpea flour you're obtaining in the US is different.
Best Answer
You can pour water and detergent inside, let it sit that way for a day to soften, then clean it by shaking the detergent solution vigorously and then filling it with water and emptying until the water comes out non-soapy. This should work, unless you let it sit for a long time empty, but with the residue of polyunsaturated oils inside. In that case, they can have gummed up.
If the brush problem comes because a straight brush can be stuck inside, but can't reach the nooks and the bottom, you could use a curved brush. These are sold to homebrewers in large sizes, I don't know if they are available for single bottles too.