The increase in premiums is to cover extra cost *AND* provide an incentive to lose wieght.
The problem with making the insurer do this is that it is more or less an invitation to "death panels". It's also inflexible and slow.
The traditional approach is to apply "sin taxes" at the point of consumption, such as duty on tobacco and alcohol. Experience shows, however, that governments like to set such duties to optimise revenue rather than reduce consumption. Such duties are evidently (we have years and years of evidence) not sufficient on their own; the reveues rarely cover the cost of care and never the loss of earnings to the employer. There is a mooted sugar tax but Coca-Cola et al. have already successfully lobbied against this. And if you think the UK and the US are bad, wait till you see Latin America!
So you need additional measures including education, including labelling, and possibly even bans, assuming you have legislation the lobbies can't sue you over. It would also be nice to reduce subsidies (both tobacco and sugar have direct subsidies, alchohol indirect). Even then you'll have to fight tooth and nail: the public health arguments against tobacco were no longer disputed by the 1970s and it took nearly 40 years to get the legislation that made a significant reduction in the number of smokers.
But at the end of the day you're also going to have to live with and budget for a part of the population with unhealthy life styles. Paying them to go for a 2 km walk everyday is about the best you can hope for.