Where are you located? I had this same scenario and it took over six weeks to get reimbursed. I thought they had since signed a national contract with Hertz though that would allow Hertz to direct bill Lucid.
Those that live close to Service Centers appear to have an easier time on many levels, including the sourcing, delivery and billing of rentals provided while vehicles are being serviced by Lucid.
I really believed, based on discussions pre-delivery with my DA as well as his supervisor and another higher level Lucid person, that there would be additional Service Centers much closer to me in the near future at the time of discussion. Those discussions were last March/April. Those comments appear to have been made just to appease and address my concerns. I consider one of those conversations (one with the higher level person at Lucid) to have been blatantly misleading in hindsight. He led me to believe that an arrangement had been made with a shop closer to me that could service Lucid vehicles. What he did not elaborate on was that the shop was a body shop for damage repairs and not a shop to address warranty or other issues, my fault for not asking more questions, I suppose. I still have those concerns, almost a year later, and there have been no new announcements of "new Service Centers coming soon" to my area.
Living in Charlotte, over 600 miles to the closest Service Center(400+ miles when the location in McLean, VA opens). I have had visits from Mobile Service from both Atlanta and Richmond and there have been times where my service requests have been "lost" between the two as there is not common management of those capabilities. I believe the Atlanta team is run out of Riviera Beach and the Richmond team is run out of somewhere else.
