The procedure is not available without DME regardless of how one is positioned onto the LOC.
The easiest way to see why this is the case is to look at the NDBDME28 procedure and then at the RADAR NDB28 procedure.
While one could think that radar could fix the distance from touchdown as accurately as the DME, but this is very dependent on the radar performance, how the procedure has been designed and if radar fix was considered at the design and approval stage.
You will see from looking at both procedures that the minima for the Radar NDB (which you might expect to be the same as the NDBDME) are the same as the crossing height of the 3DME step-down fix height on the NDBDME.
Without having access to the design info, one would expect that the radar performance is such that it can not be used to provide fixes close in or that this was simply not checked.
Even the last fix on the SRA is at 2nm with the pilot having to time the final part to the MAP which is at 1nm.
The ILS has a close in check fix at 0.9 DME which is also the MAP for the GS out approach. Some clutter or other issues could prevent this fix being accurately provided by radar.
To me everything points to poor radar performance close in.
As for using GPS. Remember that the DME reads zero at the threshold. Unless set for the threshold, the GPS will often read zero at the ARP.
Regards,
DFC