Dog Fight / Compensating other party

My husband was walking our dog yesterday. He typically walks the dog on a 6 foot leash that he ties around his back so the leash ends up being about 4 foot long where our dog is within arms reach. He holds him this way because he is a strong big dog and he pulls sometimes. While my husband was walking the dog a small, off-leash Dog approached ours at a came around a corner at a running pace so my husband immediately tried to pull our dog out of the way and turn around. The small dog persisted and was unrestrained so it jumped in my dogs face and my husband tried to turn around and pull our dog out of the way but when he turned around to scoop our dog up to avoid an altercation our dog had bitten the other dog and my Husband spent I’m sure what felt like an eternity trying to separate the two dogs. The other dog was injured and the individual who had been walking the dog, and his girlfriend (who was the actual owner I believe) Came out to get the dog so they could take it to the vet to seek care. My husband told the man who had been walking with the small dog how sorry he was that this had happened and the man said “it’s not your fault, it’s mine. The dog wasn’t on a leash”. After a few hours someone who I thought was owner called to let us know that the dog had been euthanized and that upon the vets advice They had filed a police report. We felt terrible. We called the police/animal control of our city (right outside Chicago) to let them know our half of the story for the police report. They came to our house to do the normal protocol when there is a dog bite such as taking a picture of our dog and giving us paperwork to sign and for instructions to follow and they let us know that the other individuals would file the report had claimed that their dog was also on a leash at the time of the incident, which was a flat out lie. A few hours after that a different girl called who I guess actually was the owner and the first person who called was a friend because she might’ve must’ve been too distraught, saying she wanted us to help pay for the dog being put down because she couldn’t afford it (The vet had recommended some treatments and surgeries to help with the damage caused to their dog but she could not afford them because vet care is ridiculously expensive so instead elected to have her dog put down) And we were responsible as our dog had caused the injury. I was furious and didn’t want to help pay because she had lied to the police about Their dog being on restrained in the situation, and they should not have had their dog in an unprotected situation. Had she been honest with the police I would be totally willing to help her out because the situation is harrowing and terrible for her.

Do we contact a lawyer? What kind of lawyer if so? I was willing to help her with some of the costs of the euthanization As a gesture of good faith towards someone who just lost their dog in a horrible situation but not as an admission of guilt. However now that she has chosen to lie to the police about her half of the situation I am unsure if I should help her out as she is seemingly are now untrustworthy. Any help is greatly appreciated.