Thing is, your daughter does not have a choice about whether or not to come along on your ride. Your life is her life and she is an integral part of it.
Your daughter is number one. Children have the innate right to be number one in their parents lives. You set the example for her in what is acceptable or not acceptable in life. If you and your gf have a falling out, you can try to shield your daughter all you want, but she will be affected by it no matter what.
Your daughter's mother is looking out for her daughter. Growing up in an alcoholic household can seriously impair her emotional development and it sounds like her mother wants her in a stable environment - which again, all children deserve. It is their right.
Now is not the time to consider yourself or your gf first. Now is the time to recognize your daughter's needs and what may or may not hurt her and move forward. Your gf and you may not appreciate the 6-month rule but that rule would not have been put in place if things had not happened the way they did. There was a consequence for her actions and she should accept the consequences and realize your daughter's mother has a fair point.
You've known your gf a short time and seem to be willing to do whatever you can to make yourself a family. I understand that. But your daughter is part of the family and having a great time together does not a family make. If it was me, I would actually be curious to see whether or not this love is strong enough to survive a few months of this. If it is, then you can really sit back and tell yourself your gf was committed to this relationship.