Is there a name for the Southern verb form "done" + past tense?

Such use of done may be called the completive ‘done’ or the perfective ‘done’, an example of a completive/perfective aspect[ual] particle:

• it is a particle, i.e. a word which serves as a marker of grammatical relationships between other words rather than expressing lexical meaning in and of itself
• it indicates the completive or perfective aspect, i.e., that an action is completed, terminated, already over, etc. In standard Englishes, the perfect would be indicated by have or has.

The companion website for American English: Dialects and Variation, 3rd Edition, by Walt Wolfram & Natalie Schilling, states

Completive done
The form done when used with a past tense verb may mark a completed action or event in a way somewhat different from a simple past tense form, as in a sentence such as There was one in there that done rotted away or I done forgot what you wanted. In this use, the emphasis is on the “completive” aspect or the fact that the action has been fully completed. The done form may also add intensification to the activity, as in I done told you not to mess up. This form is typically found in Southern European American and African American vernaculars.

I don't know if there's a "name" for these parts of speech, but as a native Southern American English speaker (or at least listener), I wouldn't say that "have been" and "done been" are equivalent. Depending on emphasis and context, "done been" indicates not only that something has been completed, but that it was completed some time ago, while "done been" emphasises the completeness of the action. For example, "I done been to the store" means not only have I already gone to the store, but that task was completed some time ago. "I done been to the store" with neutral emphasis is equivalent to "I have been to the store," but "I done been to the store" is not. And "I done been to the store" would emphasize the completeness of the task (possibly implying an unwillingness to return).

As contextual examples, if Sandy asks Jeff if he's been to the store 20 minutes after his return he might reply "I done been to the store." If she asks several hours after he's already gone, he might reply "I done been to the store." And if he feels that Sandy is asking to remind him of a responsibility to go to the store and wants to emphasize that he did, in fact, compete the task that was set for him he might say "I done been to the store." He may also say this if she asks if he could get milk at the store, and he feels that she should have asked beforehand and is unwilling to return.

These are just examples from my particular pocket of the South (middle Alabama), so I don't know for sure if it's true in all SAE, but at least where I live, "done been" is not strictly equivalent to "have been."