Drafting up the expenses agreement, having the board approve it, and having the comptroller deal with it seems like a lot of work that could have been dealt with by a simple conversation instead.
I fly my own (single-engine piston) airplane for recreation, most of our east of the Mississippi family travel and have flown it for a prior company on business trips. Even my little bug-smasher can save a night's hotel stay on one end, and often on both ends, making my life and my family's life better for that, making me more willing to travel for business, and I had no problem (ethically or practically) submitting a spreadsheet with column B as what the airlines would have cost, including airport parking and hotel charges, and column C showing what my airplane cost (including hotel and landing/ramp fees) and submitting for reimbursement the lower total (almost always airline, but saving 2 nights hotel and 4 days parking at Logan takes away a good bit of the difference).