Hi - the person you hit does not have to use their insurance company. Even if it was their fault they could decide to deal with the matter themselves and as long as they paid up as required then that is fine. They should of course tell their insurer about the incident as that is a condition of all insurance policies and if they then wished to use their insurer for the incident at a later date there could be problems if they had inadvertently done something to prejudice their case. So not a desirable option but an option nevertheless.
You cannot dictate to the Third Party.
If you are worried about them making an inflated claim then tell your insurance company and explain why you think this. But, at the end of the day you or your insurer need to prove what you are saying and to do this you need evidence e.g. photos of the incident, an independant witness, or an independant engineers report on the Third Party vehicle. I am afraid that the costs of arguing your case are often higher than the cost of simply settling up - and if this is the case your insurer will probably just settle with the Third Party.
If your insurers make a payment and cannot recover it from the Third Party then you will lose part of your No CLaims Bonus regardless of how much they actually pay out. Sorry, I know it is infuriating but that is the way the system works - just take a step back and let your insurer (the insurer of the car you were driving)get on with it.