No, it's not possible - this is a requirement to be treated like a Bulgarian for taxes. Maybe be you should check if you spend 3 months here and in all other countries less than 3 months (3 is a primer - it maybe 2, 4, 5) whether it will be enough as you "lived" the longest time in the year here. This is just a guess and probably depends on your national taxation.
And yes - Bulgarian government knows if you are in the country or outside. Maybe if you come with a car you will have a chance not to mark your documents but I would not rely on this.