Doesn't adding to your maximum all up weight take you over the maximum all weight?
If it takes you "over" it doesn't mean you will fall out of sky or be unable to take off. Your performance will be gradually degraded the more you go "over" on the other side but you may still be able to fly just fine. Professional ferry pilots routinely get permission to take off with aircraft weights significantly over the MTOW. I am not advocating operating outside of published book values just explaining basic realities and physics of flying.
Although this seems obvious it is rarely stressed in most training documents.
It was very clear in my training textbooks, I am surprised not in yours.