I do apologize for this being so incredibly long, but i just wanted to add in every single detail so you would be able to answer appropriately and accurately.

I have a purebred, AKC registered Rottweiler that is four years of age. He is a professionally trained PPD (Personal Protection Dog) which i paid in total $5,800 for. I purchased him from a reputable breeder when he was eight weeks old ($3,200), both of his parents were highly valuable, excellent SchH working dogs (III). I just thought i would give you some background information before i explain the situation that i am in.

My neighbor had a four and a half pound, eleven month old Chihuahua (mix?) that managed to slither underneath the two inch aperture in a very small section of my six foot, thickset, wooden, privacy fence (They allow this Chihuahua to run loose every single day with out any supervision). I noticed the crevice the same day the situation that i am going to tell you about occurred. As soon as i noticed the hole i placed three heavy set stones over the crevice to avoid any intruders from entering my property, but some how my neighbors Chihuahua managed to alter the position of the stones and make its way on to my property. My husband was going to be repairing the crevice the next day as he didn’t have work and i agreed that it should be fine until then.
Moving on to the incident. I allowed my Rottweiler to go outside in to my backyard like i always do and i always check on him momentarily through out the time that he is outside (I would estimate it to be every five minutes or less), he usually remains in the backyard for twenty minutes or so before he returns to the door to be let inside. The first two instances that i checked on him everything seemed to be normal (I didn’t even see the Chihuahua in my backyard). With in the five minutes between the second and third time that i would be going to check on him i heard a high pitched shriek from outside, it resembled a car more than a dog, so i immediately bolted to the front yard and saw absolutely nothing. I heard the shrieking again and practically flew in to the backyard in a panic as i realized where it was coming from and my dog had a small creature in his mouth which caused me to panic even more so than i was already. I gave him three commands “Aus”, “Hier”, and “Bleib” which he obeyed instantly. To my horror it was in fact my neighbors Chihuahua. I picked the dog up right after allowing my dog back in to the house and brought the seriously injured Chihuahua over to my neighbors home which they then immediately brought the dog to the Veterinarian. I was later informed that the dog was deceased, since there were no visible wounds on the dog, i assume that there were internal injuries/bleeding.
I do take full responsibility for the incident that occurred and i am willing to pay the Veterinary bills provided. The problem is that they want $4,350 for the Veterinary bills, $900 for the dog, and $400 for pain and suffering but they will not supply me with any billing for the veterinary services, nor proof that they paid $900 for the dog. I do feel absolutely horrible about this whole situation but i am not sure that i should trust their word on this, i just don’t understand why they wouldn’t supply me with any bills/proof, it is a simple and (I believe) needed task. My neighbors keep threatening to call the police and have my dog humanely destroyed because i refuse to pay them at this point due to them refusing to supply me with any proof but they haven’t done so as of yet. Every time i have asked them for proof, they become very belligerent while repetitively screaming “We do not have to supply you with anything, but YOU have to pay us for our dog!”, i have tried repeatedly to speak to them about this but to no avail.

Should i just cut my losses and pay them? Should i allow them to just take me to court where they will have to show proof of bills/cost of the dog? Any advice is very much appreciated. Thank you.