I've only got to the third room, but it happens in the second and third room while using the rock for the, er, monster, and the key and sword respectively.
Hey Inferno, thanks for your patience.
I just tested on Chrome (Version 73) on Windows 10 (I don't have Chrome OS) and I was unable to reproduce the issue you mentioned (even if I click on the red monster when it is in front of any door).
I often use Firefox, and with Firefox on Windows 10 I never had this problem either.
Since I don't have the problem it's tricky for me to debug.
I know that there exists some other ways to publish Pico-8 games, for example by providing a binary executable which must be run on your computer. Maybe with this workaround the problem could be solved. Unfortunately I don't have any experience with this kind of Pico-8 publishing. Are you interested in testing it as an executable? Here's what I can do https://pico-8.fandom.com/wiki/Export