Short answer: In general, yes. But it is not recommended.
The longer answer:
What are considered such “management plugins”?
– WP Custom Admin Interface (free, by Martin Gibson)
– Admin Menu Editor Pro with the “Toolbar Editor” Add-On (Premium, by Janis Elsts)
As Toolbar Extras does only extend what is there from default WordPress, these management plugins should in general work pretty well with Toolbar Extras.
However, the Toolbar (Admin Bar) API in WordPress is really limited and was not intended for a huge amount of items and also not for a sophisticated, dynamic management of items (nodes). This is a fact that cannot be ignored. And that is the reason, the above mentioned third-party plugins (and other, similar ones) will not always work properly.
From what I have tested so far, it should work well together. But if you have a big install with a lot of plugins, those management tools have to load a lot of (Toolbar) items into the memory of the management interface. This can lead to timeouts and crashes, whatever… Also, once a configuration of differently organized Toolbar items is saved it could happen those items are not displaying properly as intended.
Therefore I do not really recommend the use of these or similar management plugins. If you need to remove or add Toolbar items, always do it via code. It works way more reliable – by far! And it is way more efficient. Put all these code snippets neatly organized into the Code Snippets plugin. This makes so much more sense. You will save a lot of money and headaches also. Believe me, I have been there, more than once.
Another thing, I have experienced with those plugins, they tend to be heavy on resources from time to time. It is not always joy to work with them.
That all said, please don’t get me wrong: Those plugins are incredible on their own because they bring solutions to areas where WordPress Core totally fails and is not well thought yet. However, as WordPress Core is somewhat limited still in these areas, those management plugins can do no real magic to overcome these limitations no matter how hard they try. This is my personal experience after more than ten years of trying to customize the admin area in the way that I wanted. Again, those developers and their products are top-notch. This is a fact. They achieved incredible things. I just would not recommend them to use together with Toolbar Extras. Or if you still want to do that, you are totally welcome – but please be aware of the limitations.
Important Note – Support Policy:
I will offer no support for any issues for those management plugins in regards to Toolbar Extras’ features.
That is because I will not invest money on these (premium) plugins only to test them for support. This makes no sense. Also, it is a “colliding of concepts” here. My plugin does add stuff in a smart way. Those other plugins more or less remove stuff. So you have to decide which is your higher priority. Both concepts together is way better achieved via a lean, minimalistic code-wise solution.