We stayed at Le Petit Hotel last year and Bleu Emeraude (basically next door) last month and I thought I could provide some additional input. We walked to/from the restaurants each night, and this year we did have my in-laws with us who are in their mid 80s. We never bothered with the shuttle last year. We were planning to arrange taxis for them each night (Bleu Emeraude does not have a shuttle) but they wanted to give walking a shot the first night and they were ok with it for the rest of our trip. Most of the restaurants are about 1 km from Le Petit Hotel - about 15-20 minutes walking at a leisurely pace, or about 10-15 minutes walking at our usual pace without my in-laws. They took a taxi one night back from Spiga ($10) as it's a little further away (1.5 km) but other than that they walked.
The first 400 m of the walk from the hotel to where the road splits (at Captain Frenchy) is the problem - the sidewalk is poor or very narrow in parts. Once on the one-way boulevard, the walking is easy. So on this 400 m, we just took it slowly and carefully. Some cars coming? We waited until they passed by and then proceeded a little further forward. We found the walk back a little easier because there is less traffic later at night.
But you do have the shuttle option, and it's easy to get taxis too. I think you will like Le Petit Hotel a lot, and Remy, Yamina and Myriam will take good care of you. As you know, as a guest there you can go over to Hotel L'Esplanade and use their pool area, which is certainly a nice place to relax and have lunch and drinks around the pool.
Our restaurants this trip were Spiga, Le Cottage, Bistrot Caraibes, La Villa, L'Auberge Gourmande, and Villa Royale (all excellent although Spiga and Le Cottage were my favorites). We also did a super-casual Friday pizza night takeout meal from Rotisserie/Pizzeria Les 3P which is just 200 m from Le Petit Hotel - it doesn't look like much but we had a great pizza, small rotisserie chicken and a big salad that we brought back to our hotel. I would suggest making some reservations before your trip, if you still have time. If you're there on a Tuesday night when Les Mardis de Grand Case goes, definitely make reservations for that night because it's busy. We happened to have reservations that night for 6:30pm at Bistrot Caraibes and when we were done, we walked out just in time to catch the parade.
Have a great trip!