Can I catch a bus that would take me from Luxor area of the strip to the rental car facility?

Or is the closest I could get be the south strip outlet mall area. I would hope to not have to change buses. If
the only way to get dropped right at the rental facility would require a bus change I would rather just get off at the outlet mall because if I remember correctly
it would just be a a short walk on Warm Springs to Gillespie.

You can check with the https://www.rtcsnv.com website, but for an alternate route, an option is that you could take a walk to the bus stop by Mandalay Bay going Southbound and from there take the SDX bus to the South Strip Terminal Bay 17. From there, you would then take the 109 Bus Northbound to McCarran Airport where you should be able to get to the Car Rental Facility Shuttlebuses for a Free ride from there.

But the RTCSNV website does seem to say that it is about a 10 minute walk to the Rental Car facility if you were to walk from the outlet place if you took that option.

RecVPPlayer

I had thought about getting to the airport via bus and then getting on the car rental facility shuttle to the car rental facility, but I've never quite been sure if I would be dropped in an area that would allow me to access the arriving flight/baggage claim area in order be able to jump on the free shuttle. I always just assumed that the only place a cab or bus would drop passengers would be at the departing flight area of the airport.

I wouldn't worry one bit about getting checked for proof of car rental when boarding the airport car rental shuttle bus. There actually are tonnes of cabs waiting for people and there is a drop off zone for cabs also.

I'll have my rental car paperwork. It's just that I am coming out a day earlier than I originally intended and as of now it is cheaper to keep my current rental the way it is than to cancel/modify and add another day, so I may just want to make a separate reservation to cover the 1 1/2 day that I will be arriving earlier than originally planned. Regardless of whether I make a separate reservation I will want to find a way to the rental facility.
It may sound kind of strange because I can afford a cab, at least someone going to Vegas should be able to, but as a many, many, time visitor, public transportation (bus) from the airport to my hotel and then later in the day or the next day to the rental car facility would be something I've never done.
The other option would be to just book the separate reservation for the morning I arrive and be done with it.

Yes, you can easily take a bus from Luxor to the rental car facility.

Board a southbound Strip & Downtown Express in front of Mandalay Bay. Get off at the first stop as it turns left onto Warm Springs. Walk 1.5 blocks east to Gillespie, and half a (long) block north to the rental car facility.

Easy peasy.

Thanks for the responses.