WesTX, great story and glad you've decided to return to the island with your children. Our first trip to SBH with our children was Feb 2017 - they were 3 & 2 at the time. They're now turning 7 & 8 in a few months and our November trip will be their 5th to the island (rough life).
As others have noted, most restaurants do not open until 7p. In the early years at least one of our children would fall asleep at dinner. Tamarin and Bonito do take 6:30 reservations if memory serves. Bonito also has a good children's menu (although 40 euro) and banquet seating (request) which is helpful if a little falls asleep (as one of ours did there years ago). Eddy's is also a fine choice with children (no specific children's menu) and they too have a table with a long cushioned bench (another sleeper one year for us there). Tamarin has a children's play area our children enjoy and they have a good children's menu - the setting is also magical for a child. Maya's was very accommodating when our children both (rare) fell asleep at dinner on our last night in 2017. They pushed two chairs together and the children were comfortable and cozy sleeping (see photo below).
Other more casual dinner options that are good with kids are Les Bananiers as previously mentioned and Le Bouchon. In 2017 we were introduced by way of the hotel to a wonderful babysitter who has a babysitting service on the island. We have been using her every year and are continuously impressed by her service and staff. If you would like her information please direct message me and I will pass it along. Each year we try and pick a few adult nights to dine solo (Le Esprit, Orega) and the kids have LOVED the babysitters.
For lunch there are more kid friendly options. Restaurants on the beach are always a great choice - Shellona, Lil Rock, and Sand Bar. We have yet to try the Toiny beach club but plan on doing it on the next trip. Sadly we loved Gloriette but it's no longer there - they have a wonderful casual restaurant in Gustavia but the old toes in the sand lunch was special. We also love to go to Gouverneur beach and spend lunch at Santa Fe. The lobster tank is always a visual treat! Fish Corner is a delicious meal but limited on the children's selections. Our son is a great eater so he enjoys the octopus there (he loves Santa Fe's too), our daughter on the other hand just eats french fries and ice cream that meal - oh well. Jojo burger is also a lunch favorite. Maya's to Go and Le Petit Columbe are great options for take-out lunches.
There is a lovely children's shop in St. Jean. I recommend going there to buy beach toys. They also have other great toys and games and we've purchased many favorites there. As others have mentioned, St. Jean is a great beach for kids. My daughter also loves Shell Beach - although bigger waves than St. Jean. We brought life jacket swim devices the first year ($20 on amazon) and left them at the hotel when we checked out.
I recommend a sail with Jicky Marine. Our son developed a love of sailing being on the boat each year in St. Barth. He's now going into his second summer of sailing instruction and I do believe his passion was sparked on Jicky's Cat. We've never tried the yellow submarine.
As the kids get older there are hikes, swims and different spots for jumps. Until that time, enjoy the beach, the sand, and the ocean. There is so much natural beauty on the island for the kids to enjoy. And turtles, lots of turtles.... (check out the entrance for the hike to Colombier).