Thanks for the feedback. You’re right that this might be throwing in too much complexity.
To provide context I work for a startup, and our current policy is that we can book the cheapest inflexible non-stop economy fare. We can book direct with the airline (no corporate TA/Concur).
We also don’t have an expensing mechanism (to encourage the right behaviours) so bookings must go on a company credit card.

I definitely don’t want to do something illegal or that would mess with our company’s accounting (so yes canceling a ticket and using the receipt sounds like a bad idea).

The joy of being in a small structure is that everything is more flexible within reason (so extending a work trip as a holiday would be acceptable, or starting from a different origin), but the key thing is that we shouldn’t make it more complex or expensive for the company as a result.

So I think the conclusion would be that the “right way” of upgrading a trip would be booking the economy routing and calling to amend and collect any additional fees on a personal credit card (without trying to change the trip from flight only).