Sorry, I am always confused when it comes on 12 o'clock. Is 12pm noon or midnight?
I think that's noon? If so,
there is a flight on Olympic Air departing Athens at 2pm, arriving Naxos at 2.45 on 23rd September, while there are a few seats let on the 27th on the 3.05pm flight back to Athens.
There is a 3.35pm flight Athens to Paros on the 23rd and a 4 different flights with limited availability Paros to Athens on the 27th.

Both those islands have large permanent population so they'll not feel deserted even towards the end of the season, and have plenty to keep you busy for 4-5 days. Of course you do not mention your interests, so it hard to advice. You mentioned Mykonos so I thought you might look for something on Cyclades. Actually even Mykonos might be a good idea late September. Your budget is more than enough to stay in a nice 4* hotel then.

Thanks for the response. For my time there I am looking to relax, explore, eat and drink well, and spend time on the beach / swimming. I would like to find a nice place to stay with a good view and pool that isn't too far from civilization. But walking to towns / stairs (Santorini) aren't an issue. I am also thinking of renting a motorbike / scooter since I ride at home.

Looking for a relaxing, nice place, a bit slower paced than Mykonos, that does have some nightlife

By "some nightlife" if you mean clubbing, then even Mykonos will be slower paced in late September. If you mean dinner in a restaurant or taverna, followed by drinks in a quiet bar, then I can recommend Paros or Naxos, which get very quiet after mid-September. Mykonos and Santorini, along with larger islands like Rhodes, have a longer season stretching into October.

Towards the end of September/start of October the weather on Cylades ( and most of Greece for that matter) gets somehow unpredictable, and some rain is not unheard of. It is not hot, but in general on sunny days it is still warm enough to swim and sunbath, although you may need some light jacket after sunset.
Naxos and Paros are famous for their long sandy beaches, while there are beaches attached/a few minutes walk to their main towns. Paros is more suitable to explore on a motorbike, as it is a flat island with a few hills inland and most places of interest are on the sea side. Exception to this is its former capital up on the hills called Lefkes, which is a very scenic and well-kept Cycladic village, and a few less important attractions such as Marathi marble quarries and Petaloudes valley. Paros has a very good bus system too. Parikia, the capital town of Paros features a couple of sandy beaches on a 10-20 minutes walking ratio from the harbor, while in season there are quick boat trips to other beaches near by. Parikia is a lively town with very decent nightlife even on September. It is scenic with some windmills, a castro ruins area, its Cycladic narrow streets and its church of 100 Doors. The only issue I can think that might or might not be a problem, is that very posh hotels on Paros are built on isolated locations. I am more familiar with accommodation on Naoussa which is a very scenic sea side town on Paros, and sort of its second major spot, preferred by many visitors over Parikia but I am a bit hesitant to suggest a stay there because it might be far too quiet at your dates. Maybe someone else can advice on Parikia accommodation with a pool and a view...
Naxos Town is a very scenic one, with a nice harbor, the famous portara (Ancient Gate), a very picturesque old town/castro area, a couple of very worthing museums (Venetian, Archeological etc) and St George beach area attached to it. St George is a shallow long sandy beach and there are many accommodation options immediately on the beach or a few streets backwards. This area although very busy at day, is suprisingly quiet at night, and it has a residential feel on it, with many local houses among the hotels, bougainvilleas and vineyard trees etc. It is only a 15-20 minutes walk to the harbor, depending on your exact location. You may want to have a look on Naxos Resort, Nissaki Hotel and Galaxy Hotel.
Naxos features the highest mountain of Cyclades, mountain Zas and in comparison to most of the other islands on the area, has lots of water so it is lush and fertile. Maybe its stronger feature is the inland villages where locals deal with their everyday life, unaffected by tourism. Agriculture is their main job, and sometimes you feel like time stands still when there. A motorbike might not be very suitable to explore inland due to inclines on some parts. In addition to said villages, there are many ancient ruins, kouros statues lying on fields (!), tiny very old Byzantine churches, marble and emery quarries and much more to see on Naxos. From town, moving southern there are many great sandy beaches. Local bus system will be fine to connect on 2-3 of them on late September, your own transport is advisable to explore the rest.
Depending on how much demand is there, and on weather conditions, there are boat excursion departing Paros and Naxos and taking in Mykonos/Delos or Santorini. Paros offers the near by Antiparos island on easy reach as a bonus, an island with a cute town, many sandy beaches, a deep impressive cave in the middle, and boat trips that take in sea caves on its rough side and visit inhabited near by islands. Plenty of info on this forum and other travelrs' forum about Naxos and Paros.