I am not sure whether you use 'that' as a conjunction after the noun 'statement'.
The statement [that] A is more than B is false.
I do not agree with the statement [that] John often shirks the work. This is a slander.
Are they grammatical?
The word that is optional in constructs like these:
We know more people buy red cars than pink cars.
We know that more people buy red cars than pink cars.The graph shows the price of oil has declined steadily over the past three months.
The graph shows that the price of oil has declined steadily over the past three months.Becky said she might be late.
Becky said that she might be late.
In the example you provide, there's no need to include "with the statement", although the word that must be used if we leave it out:
I don't agree that John shirks his work.
I don't agree with the statement that John shirks his work.
If we include the with the statement, we can omit the that, but we should punctuate the sentence accordingly:
I don't agree with the statement: "John shirks his work."
You could accomplish the same thing with italics rather than quotation marks:
The statement A is more than B is false.
Whether or not I include a that in such sentences comes down to personal preference and readability. Sometimes the sentence will read too "choppy" with the that omitted, and sometimes the sentence will seem too wordy with the that included. It's a case-by-case judgment call.
(shirks the work),(shirks work)
.