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