I did a bit more testing. It seems that the Display Options for Resolution & Window Mode are not being saved/loaded. This is not just when restarting the game by refreshing the page, but also when starting a run and returning to the Menu. I was trying to experiment with different settings to see if they had any affect.
I also tried switching to fullscreen as soon as the game started, with the default settings. I did not get the crashes my kid reported, though only played a couple locations. When I went back to the menu I was zoomed in at the bottom left with clouds, and just a portion of a building showing.
I'm using a mix of mouse and keyboard currently. I don't often use fullscreen for jam games, but with my extra testing, I'm acknowledging that for keyboard you heavily use Esc. On the web Esc will exit fullscreen. You should add an alternative button that can be used for web. P is common for Pause, but would not make as much sense for some of the other uses. Similarly Backspace could make sense for going back, but if the game loses focus, many browsers will go back a page when Backspace is pressed and you are generally focused on the page. If you enable both by default that could let players in fullscreen use Backspace, as they'd be unlikely to be unfocused such that it would go back in fullscreen, and those not using fullscreen could stick with the default of Esc, along with non-web being fine with the default. You could add this as a note on the page or instructional popups in the game, while leaving the default Esc in the UI. Adding input remapping would be the best to let players chose what works for them, but that would obviously take longer. A non-Esc alternative could be a good fallback for web fullscreen users for short term.