Hard to say what’s specifically going wrong here, but the dump1090 webserver is not exactly, uh, “production grade”. It’s possible you are hitting a memory leak or similar. Not sure why it would only be triggered by the kiosk’s browser, though.
At the risk of tooting my own horn you could try dump1090-mutability (here); it is set up so you can use a separate webserver on the Pi, e.g. lighttpd, to serve the map data, which is likely to be much more robust.