If your child lived in your home more than half the year, only you have the automatic right to claim the child as a dependent. There is no such thing as 50/50 custody in the eyes of the IRS, you must actually count the number of nights where the child lives. If you have custody more than half the year, claim the child and file your return. If the other person already e-filed and has blocked you, you can print your return and file by mail. The IRS will investigate and, based on the girlfriend not being married to the other parent, she would lose the claim.
If your child lived in the other parent's home more than half the nights of the year, only the other parent has the automatic right to claim the child and you can't claim anything unless the other parent gives you a signed waiver. However, the other parent's unmarried girlfriend can't claim the child unless the child lives in her home for the entire year. If the ex has no income and doesn't file a return, then no one else can claim the child as a dependent. However, for 2020, the ex could file a return with zero income and claim the child as a dependent to claim the recovery rebate for the child.
(Because 2020 was a leap year, it is possible that each parent could have custody exactly half the year, 183 days each. In that case, the parent with the higher income can claim the child as a dependent, but can't claim the child as a qualifying person for head of household, because exactly half the year is not more than half the year. In other years with 365 days, one parent must have custody more than the other, even if only 1 day more.)
However, while you might be able to punish the girlfriend by reporting the fraud, you won't get a financial benefit. You can report suspected fraud using a special form. See this link. https://www.irs.gov/individuals/how-do-you-report-suspected-tax-fraud-activity
