Is it permissible to issue fictitious invoices or invoices with data that does not match reality to cover items of a research grant, knowing that the money was actually spent on the research but on different items? What is the ruling on giving money in exchange for the invoice or changing the item of expenditure in the invoice?

What employees state verbally is not considered unless they are authorized by the granting entity.

You must adhere to the conditions of the granting entity. It is impermissible to submit fake invoices or pay bribes, as the Prophet (peace be upon him) said: "Muslims are bound by their conditions."

If the employees are authorized and fake invoices become a means to obtain your right from the grant, then there is no harm in it, such as a printer invoice if the granting entity permits its cost to be accounted for.

The bookstore owner must give you the invoice for free. If you cannot find one, then there is no harm in paying for it.