South point Circut house is the best option for you but for that one has to approch some Govt officials of that region becoz this is a Govt accomodation but worth staying
I would suggest to divide the tour 4 parts.
1. Port Blair and local sight seeing (may or may not include Mud Volcano and Baratang forest visit... personally I felt it not to be worth it, specially for couples).
2. Port Blair Island hopping trip...includes Jolly Buoy, Ross Island etc.
3. Havelock Island trip.
4. Neil Island trip. So better to book stays in three places... Port Blair, Havelock, Neil.
At Port Blair, try staying as close as possible to the Cellular Jail...coz the jetty is at walking distance...from where you will get all the ferries to go everywhere all over Andamans.
For Havelock, it's probably costlier among all the three, by courtesy of its famous Radhanagar Beach. Staying here depends on your budget. If you can spare more, you can rent Airbnbs or beach igloos at Radhanagar beach. If you want a cheaper stay, staying near Havelock jetty would be a better option.
At Neil, I would definitely recommend any hotel with private beach or beach front hotels. Tip: Try staying at Neil in a full moon night...you would remember the experience your whole life.
Take care, travel safe.