Can this work with a fonewearl?
Edit: besides the IDs you mentioned in the post, is there any other one that gets interesting things?
The quest is pretty flexible for any FO due to it only being on VH for solo play. As long as you are bringing in fire tech boosting equipment to help with clear speed, it should be fine solo.
It's also okay to have multiplayer runs of mixed FO --- just be aware the quest may be harder to clear for a group if everyone is not appropriately geared / using 29+ techs / leveled up. It can be a very big clear difference if the group is not able to clear Dorphon or Merrisa A spawns before they attack, which can happen if you aren't using 2+ FOnewman while leveling others.
When it comes to IDs:
Bluefull really outshines basically everything here, but you could in theory do these quests on a few different IDs for "okay" results. I still would not generally recommend picking another ID for these series of quests over Bluefull.
Other IDs may prefer different quests / more desert heavy alternatives to increase the odds of their preferred value hunts (ex. Zu for photon crystal, Pyro Gorans for Tyrrell's Parasol, Goran Detonator for Diska of Braveman, etc.).
If lacking ID access due to a relatively fresh start to Ephinea, see the below as possible alternatives for this series of quests.
Yellowboze - MA4B for photon crystal hunting. Alternatively you can do MA4C resets at Astarks for unsealing without backing + H/Arm drops.
Redria - MA4A for photon crystal gathering, MA4C for decent value.
Purplenum - MA4C preference, but this ID is sadly a strictly worse than Bluefull alternative at everything MA4x related (it really deserved a Vise chance here).