Did you ever by chance see the following:
One thing I have found out after working with the 3PCC phones is for some reason the phones do NOT like to dial certain * number feature codes.
I have Cisco Enterprise phones (configured with SEPMAC files) mixed in with the 3PCC phones and I have no problems dialing all of the star feature codes in FreePBX with those, but for some reason the 3PCC phones balk at certain combos. This may be related to your problem.
I don’t use ring groups or pickup groups. Instead for groups that would be similar to a “ring group” I just log a second line button appearance on the phone into the same extension. chan_pjsip allows this if you raise the max contacts. It does not behave exactly like a traditional key system does (with the flashing button lights) but this is, after all, a PBX and not a key system.