Can air-to-air refueling be done in compliance with the FAR's?
I can't provide a definite yes or no, but I can discuss some of the FARs that will need to be considered. In most cases you have to consider three groups of FARs; covering aircraft, aircrew, and operations.
For the aircraft, they have to have an airworthiness certificate.
The Kodiak is certificated, but you have to address operating without the cargo door (may be approved) and if the temporary installation of your refueling equipment needs any approvals. Many temporary installations don't but since your refueling equipment will connect to another aircraft's fuel system it may need a 337 or STC.
You say the receiving aircraft will be custom built. That implies an experimental certificate. There are two, maybe three, different experimental certificates that you could theoretically apply. The first is Experimental-Amateur Built (E-AB). This could work if you are not a business and this effort is for personal educational purposes. It's the least restrictive, but is in a gray area of applicability. The second is Experimental-R&amp;D. It would be applicable, but it's the most restrictive certificate as it limits you to just R&amp;D flights and crew proficiency flights and it requires annual renewal. Experimental-Exhibition might apply, but it's usually limited exhibition flying of existing, non-certificated (typically ex-military) aircraft. In any of these, I would expect that you would need to include aerial refueling within the operating limitations and that means getting an FAA inspector to buy off on it.
For the aircrew, I expect the one potential issue is one of compensation. If everyone holds at least a Commercial certificate that would avoid any gotchas. The other issue may be meeting the FAA training guidelines for formation flying.
Flight operations would be subject to Part 91 and the operating limitations of the aircraft. I would expect that refueling operations would be limited to airspace 'not over populated areas.'
So, I expect it would be possible but it could be difficult getting the necessary approvals.