The problem I encountered was that the key I was rewriting contradicted the mnemonics in the menu. In particular, I had Alt + i mapped to something, while it was also a mnemonic for Find.
To fix the problem, I removed the mnemonics from the menu, which stopped the menu from appearing. To do this, go to the menu "Settings" → "Browse packages" and open the menu file "Default / Main.sublime". There you can delete all the "mnemonic": "*" lines to get rid of the mnemonics.
Pressing the alt key still brings up the menu. If you want to disable this, you can use the following AutoHotkey script function (thanks Armin):
SetTitleMatchMode RegEx
LAlt & esc::return ;This can be any key, not just escape. It needed to get AutoHotkey to treat Alt as a prefix key.
~LAlt::return
Alt, , Alt .