Hi,
Your solution is correct. But the lowering of the prices should not be of 3%.
It should be : 1.09*{old price without tax}/1.06
That will give you the new price without tax.
You can actually do that all at once with a mass action via the menu System>Mass actions.
You need to have one action on the column price_value with the "operation" type and the value: 1.09*price.price_value/1.06
Once you click on the process button, it will change all the prices at once.
Note the "limitations" area inputs where you can configure how much products you want to update and a start (so that you can run it multiple times if you have a lot of products).