I'm having the same issue, only in safari. I don't mind using an override or adding something to the htaccess file, but so far nothing I do has worked to force safari to download the ics without the html at the end of the file name.
I can get around this for the event list by doing this:
$ics_link = "webcal://" . $_SERVER . "/index.php?option=com_matukio&view=ics&format=raw";
But I can't find the link for the booking page so I can modify that.
Update: found the links, they are in core helper files, but for now, I modified them till I can find a better solution.