One of the objections to my Netting IX post was that there were very few or possibly no existing helicopters capable of carrying a recovered Falcon IV stage to a safe landing even if it were possible to catch the stage in the first place. There is a somewhat tested technique for building inexpensive special purpose helocopters capable of very large payload capacities.
The technique is the autogyro that generates lift off of the forward motion of the aerial vehicle without necessarily having any power directly applied through the shaft. Helicopters use a similar technique to autorotate in after losing power. Add the Roton type tip rockets and it is possible to build an arbitrarily large lift vehicle capable of lifting very large payloads, although at the expense of very high fuel consumption. An assembly consisting of a structural truss with several large helicopter blades with tipjets could carry payloads much larger than the Falcon IX to a safe landing.
The mission technique I see would be an airplane towing the recovery chopper to the projected recovery zone before the launch without using any of the chopper onboard propellant. As the reentering stage comes in, the airplane manuevers the towed vehicle as close as safely possible before casting loose to let the chopper catch the F IX stage while using the tip rockets. After the catch, the recovery chopper climbs to a safe altitude for a reconnect with the tow plane. The tow plane drags the autogyro mode chopper back to launch site and casts it loose again for the final landing of the stage in a safe cradle.
With good planning, it would seem possible that the recovery chopper would use ten minutes or less of full power and high fuel consumption during the operation with the tow plane supplying the vast majority of the propulsion needs.
Correction, the FIV on the cartoon was a mistake that should have been FIX.