Ah man. I feel your pain. I made a few critical errors when I built mine. You say to yourself “Why did I do that? Now what?”
So here are your options. If you leave it as is, the firebrick that will be on top of the slab will get the heat sucked out of it much faster. That may not be a problem in the front where the opening is because most likely your not cooking there. But in the back where you might have fires there it will be trouble some. Not to mention the slab can get brittle over time. So food for thought there.
Or. Ugh. You cut out what you need too. You mentioned that you prefer not too. I would mark with a marker where I need to cut out. Then with a angle grinder with a diamond blade cut along thise lines hopefully reaching down to the 2 inch depth. You can use a circularsaw with a diamond blade and set the depth but your risking over running the corners. These diamond blades are a wonder. As long as you don’t have rebar they cut pretty good thur masonry products. Even then I hear they go thur rebar.
Once you have the half box cut you can then cut slots say an inch apart or two inches apart. Chiseling then should be easier. I would advise this because its very critical that you have a insulated area under you dome.
Maybe someone else’s has a better idea here but thats the best advice I can give you. All mistakes are fixable. Don’t sweat it. (No pun intended)
Hope this helps. I wish you all the luck.