In your case I'm assuming there is no intermediate Logitech program detecting/assigning these keys such as the Logitech SetPoint program (which may be another way to go, if one of the stock Logitech programs will let you re-assign the keys)
In AutoHotkey you should be able to make a hotkey to reassign it to Escape
Or reassign it to nothing
The button name may not be
Browser_Home though, depending on the keyboard setup... the best way to determine this is to open AutoHotkey once a script is running (use the icon in the tray) and then View > Key History. Press the media button and then refresh the key history to see what key it registered as--that should allow you to reassign it correctly.
To see the full list of keynames supported (such as
Launch_App1, etc.), look up the
key list topic in help.
If you can view the key history correctly but still can't get a script to remap the button properly, post the relevant excerpt of the key history as an additional comment or update to your post.