I changed the URL to use a website with a large, slow loading page, added a long-running while loop (to freeze mIRC) after the final $com() call, and then ran the alias.
The first thing I noticed is that I can type into the status window editbox during the slow $com() call. The moment the $com() call returns, the while loop runs and freezes mIRC. This seems to indicate that the COM object itself is causing Windows messages to be dispatched to mIRC while the COM object is waiting for the website to reply. As the mIRC /timer uses Windows messages, it is being triggered.