Loosing? Learn the English language, people!

Neither is a true imperative, though, but I don't think that's the issue at hand, here.

It isn't a true imperative because the suggestive voice of 'why don't you?' doesn't permit a full imperative. It's an attempt to be direct while being indirect. Nonetheless, its justification depends on the second person being addressed.

Be honest!
Do be honest!
Why don't you be honest?

The auxiliary verb is a strengthener, which is a function that becomes confused with the other normal use of 'to do' as soon as you switch to the third person. The copula doesn't take an auxiliary verb except for this strengthening function.

Looking at the very limited data above, it's to do with how dynamic the action of "to be" is.

That's certainly a way of looking at it, in view of parallel constructions ('why doesn't she take out the rubbish?') but I don't think it's the crucial issue. I'm sure that some people understand 'to be honest' as very dynamic, but grammatically it is a simple attributive construction.

A lot of native English speakers would have no problem with "why doesn’t she be honest with the British people..." It didn't raise any problems with me on first reading.

Arguably, this is precisely the general problem that keeps this thread alive. :)

"Why isn't she honest with the British people..." has a subtly different meaning here.

Which is? I don't think there's any difference in meaning at all.

However (and this may be related to how you think of a difference in meaning) I've also just worked out (I think) where this comes from. It must come out of parliamentary language, in which speakers don't address one another directly; the speaker here was an MP. I could imagine they're quite used to modifying second-person constructions in this way.