Honestly I'd just lose the dummy now if it's already causing her problems with sleep. At such a young age it will be easier to do it now than later.
You mentioned a few times about 'scheduled' feed and sleep times. At 6 weeks old her needs are going to be constantly changing, so don't stress too much about trying to get her to stick to a set time, especially for feeds. The needs of a newborn change on a daily basis, they are totally unpredictable lol
You also mentioned that you have low supply. If you're not planning on continuing breastfeeding just ignore this but if you would like to keep going with it, feeding on demand would help get your supply up. Scheduled feeding is probably one of the biggest causes of supply problems. Just something to think about