IMvHO. This issue is not who's kids are whose nor is it about 'private time'. More to the point, it is about the parents ability to set the tone of family life within your home. Do not, under any circumstance, allow an outsider to dictate the tone with their own dysfuntionality.
I am not against the so-called 'private time' and it does not matter rather they are boys or girls. The main endeavor for you and
DH is to limit and structure ... before hand as much as possible ... the 'private time' with the emphasis on blending with all the activities and 'working and playing well with others'. With the example given, you give way to lying, manipulation, abrupt changes in personality, etc etc.
These are little girls ... age 11... not 17, 23, 36, 49, 58, 65, .....
Screw the Barbie dolls ! I thought your DH handled the situation delicately. I may have kindly sent the little bitch packing.