If I may suggest, read the post (written by the webmaster) at the top of this Portugal forum, titled "Azores Islands Travel Advice-2024". There is a link in it to my Azores guide, and the first 52 pages are FREE to read. This will help with your logistical planning. There is NO obligation to purchase my 533 page guidebook.
I encourage you to allot the 10 days. You will not regret this decision. You could visit one more island, but then you might end up with flight delays or cancellations. Sao Miguel has everything you are looking for, per your written wants. Flores is quite a distance away. Save it, and the nearby island of Corvo, for a future visit. IF you really want to visit another island on this trip, take a 15 minute flight to Santa Maria. That tiny island could keep you enthralled for up to a week if you covet slow travel!
I have not stayed at Senhora da Rosa, but did drive by it in late November (2025) to research Ermida de Nossa Senhora de Lourdes, next door, and some other ermidas, just minutes away by car. There is nothing to see or visit within walking distance of this hotel, so be aware. I always suggest to first time visitors to stay in the downtown area of Ponta Delgada. There are countless hotels of great quality there. Dozens of restaurants, and plenty of interesting sites to see, are located within this compact area. All are reachable by walking. The highway is also right there, to springboard you to wherever you want to explore, quickly and efficiently.
With 10 days, I would split the lodgings between Ponta Delgada, the Furnas/Povoacao area, and maybe Ribeira Grande. You could also stay in the Lagoa/Vila Franca do Campo area. Driving distances can be long, but almost always include stunning views.
If you are a beach lover, then September should be gorgeous. I never think the ocean water is warm enough, but many locals and visitors do. The tidal pools and beaches are located all over the island. Lifeguards should be working until mid-September, and almost all of these locations have bathroom and changing facilities.
I could keep you totally busy in Sao Miguel every day, from sunrise to sunset...and past midnight :-) . This goes for all nine Azores islands. Being too busy depends on your travel style, but you must indulge in a thermal pool to melt away any stress or tension. I cherish walking and hiking in the Azores, and a car will expand your reach.
I am always a private message away for specific Azores questions. Or, just post your questions and thoughts here, so others might get answers and ideas for their visits.
Enjoy the planning!