St. Lucia might work well.
Anse Chastanet seems to have a good rep. (I'm too cheap to stay there), and their website can show you some photos of St. Lucia.
I love Bonaire, but much of it's flat, much of the hilly region is shrubby with more thorns than you likely knew there were in the world, it's arid and I don't think it's what you want this time. For a place half again the size with more civilization and topside attractions, and some of the same benefits, take a look at Curacao. That said, it's arid, and while it has some mountainous area from what I understand, if you stay 'near the action' in Willemstad, you won't be gazing at a mountain range from your hotel room. But Curacao rocks. Edit: I see from his profile the original poster has been there.
Curacao Trip Research Notes
Curacao Trip Report with SB Surge Jan. 2019
Raja Ampat 10-Night Indo-Siren Liveaboard Dec. 2 – 12, 2022 Trip Report -
Trip Report - Raja Ampat 10-Night Indo-Siren Liveaboard Dec. 2 – 12, 2022 Trip Report
I know you're not in the market for a liveaboard, but it was my one trip to the region, and I tried to infuse the report with the info. a fellow 1st timer would need.
Look, the plane travel can be brutally long, and if the word 'budget' is in your vocabulary upgrading to first class may not be in your wallet. Unless these 4 kids in particular are well-traveling and accustomed to flights, I'd think long and hard before going with this.
Those do come to mind; I haven't visited either. Of the 2, from what I recall reading, Saba is rather 'quiet' topside. 6 People looking for good times without a strong diving focus might find it a bit sleepy.Per
Travel+Leisure:
"Saba doesn’t have large resorts, casinos, fast-food franchises, or duty-free shopping malls. And there are no concrete high-rises on this island that bills itself as the "Unspoiled Queen of the Caribbean." Instead, charming, white, wooden cottages with forest-green shutters and red-tin roofs punctuate the island’s hills. Saba is also a Caribbean anomaly, the tip of a now-dormant underwater volcano with no beaches except for the tiny black-sand "disappearing beach" at Well's Bay, seen only from June through October. So, a stay here isn’t for your everyday sun- and sand-seeking visitor. Instead, Saba attracts adventurous travelers (about 9,000 a year in pre-pandemic years), who come mostly to hike, dive, or simply relax."
Dominica is much larger than Saba. It's also poor, from what I recall, with a rep. for scenic tropical rainforest hiking but I recall it being said to be a poor (economically) island. I don't know how it would be for 2 honeymooners and 4 kids (1 a non-diver)? Perhaps ask
@agilis
St. Lucia is mountainous, has some nice rainforest hiking, has Sandals Resorts (not an option with children, but gives you some idea the crowd St. Lucia draws, plus it's a cruise port), and has nice diving.
Puerto Rico might have something to offer. From what I recall of El Yunque Rainforest, it looked to be. Some quick Googling suggests it has mountains.
St. Croix might work.
St. Croix Research Report Aug. 2017 -
Trip Report - St. Croix Research Report Aug. 2017 Since I was there, a hurricane came through and which dive shop operates where has changed (I also heard they got a zip line operator), but the broad strokes overview should be much the same.
Disclaimer: I don't know just how big or numerous the 'mountains' must be. 'Mountainous' Caribbean may not impress someone from Montana.