Most garlic here in Thailand is planted in our autumn, harvested in January /depending on where you are, in my area Lopburi ,we have a big festival near her in February, known for selling garlic and the red shallots onions. often just harvested.
One problem with Thailand is that they are very conservative with a big capital C, meaning that like the same things all the time, as I said the white garlic is mostly used, the garlic you want not so much is grown ,finding growers that grows it might not be easy ,your best way would be contract growing ?,finding farmers willing to grow the crop.
The big problem would be as you said, is the soil chemistry or lack of, Thai soil is not that fertile, soil organic matter being the main biggest problem but as most garlic is grown on 2-3 rie ? plots that could be overcome, the carrot on the stick, would be saying to the farmers you would pay them a bit over the odds to grow your crop, but you would have to know about growing cost etc. to work a budget out.
Have a word with AN's drtreelove, he is our soil expert, and he lives up in Chang Mie.