AI system deemed us at fault, not true. Now no one is willing to cover

My husband got into an accident two weeks ago on Saturday. The other driver pulled out of a parking lot speeding and expected everyone to yield for him, my husband was going straight driving the speed limit and was hit.

A police report was filed and it was very clear who was at fault. Apparently GEICO has no one on the phones on the weekend so we went into the app to report the accident. No where in the app did it ask if it was at fault or not. We uploaded the info, police report and thought it was done.

Monday morning comes and we get notified that the automated system deemed us at fault. We called GEICO and the person on the phone said she’d put everything on hold until the police report is in the database. On Wednesday we get notified that the police report was finally uploaded into the police database- called Geico to only be told that they already gave the other person a loaner and was already FIXING his car on OUR insurance?? Once they realized we weren’t at fault it was a game of phone tag everyday.

After calling twice a day everyday they finally took away the guys loaner and stopped fixing his car. edited to note that at this point the other driver tried to lie saying we hit his parked car when police report shows otherwise.
Now the other insurance company refuses to take accountability because GEICO admitted fault already. Geico says there’s nothing we can do except pay out our deductible to fix it ourselves or try to call the other insurance company OURSELVES to convince them to fix our car.

What are the options here other than continuing to call two different insurances who won’t take any accountability?