Travel reimbursement and "bonus programs"
Unless the funder has a specific policy (which they should tell you about), common practice is that "reward points" (including frequent flyer miles, etc) are yours to keep, and don't deduct from your reimbursement.

You're right that there is something a little shady about this, but basically, nobody has figured out a better way to handle it, and travelers are used to keeping their points and would get mad if they weren't allowed to.