As a "happy medium" between rail-friendly Isaac Walton and mom-friendly Whitefish, consider staying at one of the Glacier Park lodges. Several of them are only a few yards from their respective train stations. Then, you'd also have access to the park's bus system. The lodges may be slightly more expensive than staying in Whitefish (depending on what sort of accommodations you're looking for), but you wouldn't have the expense (and environmental impact) of the rental car.
I've been to the Whitefish area several times, and there are some nice places there. It's a pretty area. However, if I could go there again, I would definitely stay in one of the park lodges, partly due to the reasons I mentioned above. However, my MAIN reason for staying in the lodges is that they're absolutely AMAZING. Glacier Park Lodge is probably the most impressive, and only a short stroll across their lawn from the Amtrak station. Check out their website
here. I'd also recommend the Lake McDonald Lodge (which also has cabins right on the lake), or the Many Glacier Hotel (which is also situated on a beautiful mountain lake).