What follows is pure conjecture on who removed the pages and why.
1. The War Office removed the pages after the "diary" was submitted to them, because
a) they were kept as the hand-written signed claim, to support the transcribed text. The diary was then returned to Ashley, as he requested
b) the War Office wanted to protect officers in the chain of command from allegations of carelessness or favouritism.
There may have been financial implications, but removing the pages would have no effect on the judgment.
2. Lawyers handling the deceased estate removed the pages before offering the diary for sale, because
a) they were nervous of being sued if the claims weren't upheld, by someone named in the missing pages
b) to protect Ashley's memory if the claims were seen as delusional. There was a Consular official John Ashley who died in a mental hospital. There's no way of knowing if it's the same John Ashley. If the claims were rejected, perhaps even the family had doubts.
3. Some-one had a conflict of interest in the sale of the medals.
John Payne's medals sold for £4,000 - four times the estimate. Any doubts about the DSM could affect their value.
Both the medals and the diary came on to the open market within a few years of each other. One had the potential to harm the other.
This is the least likely scenario - simply because removing the pages didn't remove the underlying claim that 'a very great mistake' had been made. Whoever tore the page in half was careful to respect John Ashley's signature, leaving the clues behind.