Would it be reasonable and legal to use AFCI receptacles next to a panel instead of AFCI breakers?
The 4x4 box right next to the service panel is a great idea, and I do that myself a lot.  

Nobody cares which wiring method you use downstream of the AFCI.  

Between the service panel and the AFCI, the inherent problem is the AFCI would never* detect a problem on that stretch of wire.  That is a vulnerability, so that is where they require metal jacketed cable before the AFCI if the AFCI is not a breaker. 

So, use a better wiring method.  I have to say, for wiring between a panel and a very nearby box, EMT is faraway the easiest way to do that IMO. Keep it shorter than 2 feet, whcich eliminates the wire derate for 5 or more circuits.  Ideally use 3/4 conduit, which has room for more than 10 wires and will give easier pulling.  

Go to a metal 4x4 x 2-1/4 box, and use a 2-GFCI lid with 1/2" of lift.  That will give you plenty of practical room to fit wires.  If needed use an EMT nipple to chain to another 4x4 box.  An arc fault inside this heavy metal is very unlikely to successfully  start a fire. 

Use THHN wire on through the conduit.  If the circuit can't easily be moved to terminate in the box, come back into the service panel with THHN and splice there.  This out-and-back-in method has 4 wires in the conduit per circuit, so watch your conduit fills.  

* Actually, AFCIs do have some ability to detect arc faults immediately before them.  We recently had a Siemens AFCI breaker faulting because it was clipped into a GE panel, and the clip/bus was not quite compatible.