without more context about what "other key visual elements" your users might want to see while that menu is open, I'll (have to 😉) make some assumptions that I base my suggestion on.
Truncating menu labels is problematic for several reasons. For users, it increases the cognitive load required for making a selection from the menu: They have to learn and remember the work-around for displaying the full labels. They also have to keep the meaning of truncated menu items in short-term memory to compare with the other menu options. Especially for users with neurodivergence that affect short-term memory (such as ADHD), this is very challenging and, thus, not very user-friendly.

This approach also increases developer effort, as they need to implement the truncation algorithm and the work-around interaction.
Hence, what I'd suggest to do instead is to limit the length of text that users can enter to configure the custom menu item labels.
That might be frustrating while defining the labels, but it avoids recurring frustration every time users access the menu. And if you define the allowed label length via user research, you can reasonably assume that that length will work for the majority of use cases.
Let us know what you end up with, and how it tests with your users. 👍