Dog bite cases create serious financial exposure. Since you have insurance and have reported the matter to your carrier, let your insurance company handle it. Your carrier, not you, will determine what they are wiling to pay to resolve the matter and will pay for your defense if necessary. What is required to prove non-economic damages in the case where a child is "house bound for 5 months and has a fear of dogs"? IMO, expert psychiatric testimony.
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Non economic damages can include emotional distress, disfigurement (I realize you said 2 cm) and can be proven by doctors who will assess the child's emotional damages (and physical if any). Counsel Daymude has given a great answer but I would add that as long as the case is settled in policy limits the insurance company, they will be able to do so to fulfill their "contract" with you. Insurance defense counsel are generally very diligent (may I say "dogmatic") and they will not just pay out a lottery or jackpot as you say unless there is evidence. Some of what you have related, being housebound, can be disproved by written and testimonial evidence (school records, neighbors who can testify to the child out playing) and if you know of any such evidence do pass it on to your adjuster who works closely with the insurance defense counsel.
It is unfortunate to hear when individuals try to milk a claim, affecting both a homeowners claim history and the cost of insurance for the public at large. I hope this answer provides some help and best of luck to you.
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