unless the laws of colorado have changed, they are a state that recognize common law marriages. if so, then a judge is not going to think much about it. however, in that law, the parties must go to the local court house and make the relationship a common law marriage. common law spouses get the same benefits as a normal spouse, the difference is "no divorce" however, they would have to go back to the court house and disolve the common law marriage. i know this because i work in insurance for employee benefits. and some of our employees live in co. and are common law married.
you need to check with your local court house to see if the law is still in effect. ask for the marriage lisc. they should be able to answer.
you can refuse, IF YOU WANT TO to let the children visit while the girlfriend is there, that IS YOUR CHOICE. but, it may cause more problems. i would suggest, very respectfully, that you all try to get along for the sake of the children. which you are doing i am sure.
as far as custody goes in your state, while calling the court house about the common law marriage, check to see if they have a law library, if so go and find about how the custody laws work, also, you can see if they have a web site and check for the state statutes. also,i have to ask this, are her children in counciling for the abuse? if so, perhaps you would like to have your kids learn to help her kids deal with it, since they may be seeing one another often. it would be nice if they all got along. i am sure it is a great change on everyone involved. keep your head up, it will all work out.
you also need to read what the divorce says about custody and visitation. if it needs to be modified then you can do so yourself. you may want to seek legal aide if you qualify.
good luck with what ever you decide to do.
this is not legal advise. check the laws of your state.