Artiem Audax

I think the Artiem Audax may be a winner. It's in Cala Galadana.

Info on Cala Galdana here. There's a sizable beach and some bar/restaurant options:

http://www.calamenorca.com/menorca-resorts/cala-galdana

Location of Cala Galdana. It's 45 minutes by bus to Mahon and costs about 45 Euro from the airport (each way). The town as a whole is similar to Punta Prima and Cala n Bosch. It's family-friendly and known, so it's not super remote.

Cala Galdana

 Artiem Audax is Adults Only. It has a spa, a pool, and a fitness center. It's a 500 m walk to the west end of beach. That's about a 1/4 mile, or 5 minutes. Total cost is $875 for the standard room with a terrace. We can book through Chase with the points we have.

We can rent bikes or kayaks from them. We can book spas or massages. There are a couple more remote beaches west - Cala Macarella and Cala De La Macarelleta - and to the east - Cala Mitjana. From what I can tell we could walk or bike to these along those trails. Might take 30 minutes to walk, but from what I can tell, the parking lots for these beaches are 15 minutes away from the beach, so you'd have to walk to get there anyway.

We could take a 45 minute bus ride to Mahon (one bus line). Or we take a 20 minute ride to Ferreries. From there we could switch to a different bus line and go into Ciutadella or we could walk 10 minutes to Sant Patrici Winery. They have wine and cheese, so we might never have to leave.