I've often found that if there is another column where I'd like to have the ECB menu appear, it's normally because it's the primary column for that particular list.
If that's the case, changing the 'Column Name' (Display name) of your 'Title' field works well. You do this via the following steps:
- Go to your List Settings
- Click 'Title' under Columns
- Change 'Column name' to your desired name
In this scenario, you don't create your other column, as you're just repurposing your 'Title' field to reflect the column you associate as the primary identifier of your list items.
This is a good way to re-purpose the built-in functionality you get with the 'Title' field, while still allowing you to display the column with your unique name. Also, you won't have to make any changes to views, etc... in order to take advantage of the built-in functionality that the 'Title' field offers.
This may not meet your needs each time, but I thought I'd share this answer, since it often will do the trick.