First, there is no such thing as "married filing as single." You can file as married filing separately. If you provide care in your home for children who live in your home more than half the year, and you claim them as dependents, you may qualify to file as head of household.
If the children live in your home more than half the nights of the year, you can claim them as qualifying child dependents, and they will allow you to claim the dependent care credit or use a dependent care FSA. However, if the children live in your home less than 183 nights, they are not qualifying children for the dependent care credit or the FSA, even if the other parent gives you a signed waiver allowing you to claim them as dependents.
Normally, I would say ignore the payment to your live-in girlfriend to watch your own kids. However, if you used FSA funds and/or want to claim the dependent care credit, you must give the IRS the name, address and SSN of the caregiver, and they will try to match the income on that person's tax return. So in this case, you would need to treat this person as a household employee. Because the amount is less than $2700, you don't have to pay social security and medicare tax, but you must pay federal unemployment tax because the amount is more than $1000.
https://www.irs.gov/publications/p926
You are not required to issue a W-2 because the amount is less than $2700, unless you withheld income tax. You do need to include schedule H on your tax return.
If you do not issue a W-2, your girlfriend should voluntarily declare the "household employee wages not on a W-2", which is reported on form 1040 line 1b.
