The problem doing that, you tackling the problem "after" its happened. Not to mention there is health conditions that affect BMI.
How do you know people with high BMI are not just having takeaways once a week?
But you did say remove all taxes which means you not discriminating against supermarket food vs takeaways which is something I am ok with. My issue really is the bigger culprit is getting less targeted.
Not feasible in practice, but a tax on energy content, offset by fibre content, would do wonders for the health of the nation.