Some interesting ideas and liberal understanding of facts.
As a pest controller myself I would obviously recommend you seek professional help with this.
The 'best' or at least most practical solution is to use second generation anti-coagulant rodenticides. There is a 'risk' of secondary poisoning to your dog - but this risk is very very low. Firstly, once a lethal dose has been consumed, the rats are 95% + likely to die in their nests away from your dog and other wildlife which may predate upon them. Secondly, the LD50 (lethal dose required to kill a rat) is many many times lower than even the smallest of dogs - your pet is quite unlikely to even eat the rat, let alone find enough corpses to even consume enough rodenticide to have any impact upon its health.
Given your location you must accept the presence of rats and other vermin, and seek only maintain an acceptable level of infestation. Look to reduce likely food sources such as compost heaps, open bins etc, ask your neighbours if they have any problems, your garden will not be the only place these pests are.
As the gentleman before me posted, I am available for free advice...
[email protected]
Best of luck