Heather, Your D12 relies on you to be her constant companion because you are safe and familiar. She's not going to go out on a limb and meet someone. I think the art class is a perfect way for her to meet other children her own age. There's always dance class, but what are her other interests? You have to remember that she's been use to having other children in the home for a number of years and her attention was on them. Now, the house is quiet and there aren't other children around, so she's looking for not only the "white noise" but others to interact with. Hobbies are great and if you check around your area, you may find some other activities that she may be interested in.
In today's society, both parents almost have to work outside the home. The reality in your situation is that you have no choice but to work and ensure that your children have a safe and stable home life. As long as you have activities lined up for your D12, i.e. school work, as well as other activities, spending time w/her in the evening and doing things w/her on the weekends, both of you should be okay. The young lady you have coming to your home will be good company for your D12 as well. The time w/her will give her something to think about and look forward to the visits.
You are not pushing your daughter aside. You have a job to work and during that time, your focus has to be on work. Check in w/her on during breaks and lunch times. Once you are home, your focus will be on you and your D12. It's going to work out. This is a new learning curve for both of you.
Just keep in mind, pushing your child away is not having much, if anything. to do w/them. You aren't doing that. What you are doing is entirely different because you are a caring and loving mother who is trying to find a balance for both your D12 and yourself. Never doubt what you are doing.