First, thanks for find this!
The problem is that while the tag is pulled out of the props during the destructuring, it is later assigned back into the props when nav is true
The fix here is to not set tag in/on props when nav is true, but instead to change the Tag variable which was defined in the destructuring. This means that the destructuring assignment cannot be to a const variable.

EDIT
Actually, I see the intent here. It was to make the Button have a tag of 'a'. In this case, there should be an aditional check that the Tag has not changed and is the default, Button before setting the tag which will be passed.
To break it down, setting tag on DropdownToggle will with nav set to true can cause issues.

I see what you mean. The easy fix would be to change const to let on props de-structure or evaulate the tag prop again.

Would that be preferred to evaluating the tag prop in componentWillMount method?

I see you are already fixing the issue. Should I stop on my pull request?

Yes, you can close this. The change was simple (1 liner + a test to ensure there is no regression) after further analyzing the code and issue to determine the intent of setting the tag that way.

fix released in 3.9.0