Your itinerary is structured well enough for the must-visits you have in mind, but is packed too ambitiously, I fear. A day trip to Mysore will be spent mostly on the highway between the two cities; the traffic around Mandya is particularly bad, all days of the week. Make sure you leave very early, or take a train. Even better, leave on the 10th and return on the 11th. If you do end up going by road, visit Srirangapatna en route, chomp on Maddur’s famed vadais, and shop for hand-painted wooden toys at Channapatna. I’m not sure how you are going to manage it alongside the longish road journeys, but you can visit the quiet Padmanabhapuram Palace from Kanyakumari, Dhanushkodi’s mesmerising seascapes from Rameswaram, and the Thirumalai Nayak Palace and Gandhi Museum (housed in Rani Mangammal’s 300-year-old palace) in Madurai. Thiruparankundram and Pazhamudircholai, two of the six legendary Arupadai Veedu temples to Karthikeya in Tamil Nadu, are also on the outskirts of Madurai. Tirunelveli’s magnificent Nellaiappar temple falls en route from Kanyakumari to Rameswaram, and the original Tirunelveli halwa is still to be found at the Iruttu Kadai (the ‘dark shop’, because it was dimly lit and opened only in the evenings) on East Mada Street.
As for shopping, the colourful clutter of vendors by the temples’ mada veedhis (chariot car streets) and outermost parikramas are fun to browse for local curios that range from traditional utensils to golu dolls. Tie-and-dye sungudi cotton saris are great buys from Madurai. Note that temples in Tamil Nadu close in the afternoon, usually between noon and 4pm.
Assuming you are vegetarian, pre-order a yelai sappadu (banana leaf meal) or tiffin at a nameless home-run mess near the Sringeri Mutt in Rameswaram (there are a couple of them). The food is delicious, hygienic, inexpensive, and eaten sitting on the floor in the family’s living room. Madurai’s countless ‘tiffin through the day’ restaurants are generally reliable (like Meenakshi Bhavan on Town Hall Road, Arya Bhavan on TPK Road, and Murugan Idli Kadai at the Melamaasi Veedhi: the last is particularly notable for soft idlis served with a variety of chutneys, and the somewhat self-explanatory jigarthanda).
In Bengaluru, don’t miss a walk in Lalbagh and follow it up with a sumptuous and delicious meal at the old Mavalli Tiffin Rooms nearby. The government-run Cauvery Handicrafts Emporium on MG Road’s vibrant shopper’s haven is surely worth a visit.
Eat-on-the-go darshinis that dot the city are a great introduction to the way Bengaluru eats (most often). Karnataka Tourism’s Bengaluru city sightseeing tours (www.karnatakaholidays.net) may be your best bet to cover the city’s highlights in the time you have.