Müller Lümmer frühstückt schüsselweise grünes Gemüse.
What does the bold word within the tongue twister mean? I cannot find the meaning in any dictionary.
It means that he eats large portions of vegetable. A "Schüssel" is a bowl, bigger than an ordinary dish. I do not think one should take it in the literal sense that he actually fills some bowls and eats the vegetable contained in them, but we should understand it in the sense of "very much".
The ending weise occurs also in "massenweise" which could be alternatively be used:
Müller Lümmer frühstückt massenweise grünes Gemüse.
See also here.
It's a combination of the suffix -weise, which can be used to form an indication of quantity, and Schüssel, which simply means bowl.
So literally schüsselweise means he eats some number of bowls of green vegetables.
In a recipe, for example, you might read man gebe das Mehl löffelweise hinzu, meaning add the flour spoon by spoon.
In another variation there is zentnerweise, which simply means a lot, and comes from Zentner, an old unit of weight.