my understanding on this

clearly the prop management company is at fault, if i were you I would sue them at the same time answering lawsuit from tenant. when tenant wants to settle you could get prop management company to settle with them for $10k. they should have charged non-refundable pet deposit at the beginning instead of taking it out of security deposit, especially the tenant already put on checklist that there were stains on carpet when they moved in( now looks like they know the rule quite well ). how did you lose $4000 in total, did you actually settled with their insurance company?