Yes reflux is a common and well known side effect of progestogens.
Progesterone is the pregnancy hormone, makes everything slack and loose in prep for giving birth. Unfortunately this includes the valve that keeps the acid in your stomach.
Firstly I wouldn't double up on progestogens.
If you need birth control or bleed control and are taking desogestrel anyway, you can just take two of them daily (ie150mcg) and drop the micronised progesterone. This is unlicensed but supported by the BMS for endometrial protection.
If you don't need birth control, think you are postmenopausal and would prefer to use micronised progesterone then I would keep this and ditch the desogestrel.
You may get some transient withdrawal bleeding but this is not the same as your periods coming back and should quickly settle.
If not you can always restart the desogestrel.