Another detail that I don't think has been brought up here: I think I ran into a tweet that explained that the multiattack move on various monsters is their special way of reflecting the multiple attacks class features that character classes get. As such you're not supposed to be able to use it multiple times on your turn.
yep multiattack is an Action, not a "melee attack"

Mind you, though, that if you have the means to double your actions (e.g Fighter's Action Surge) you *are* able to redo a multiattack!
You threat it as if it was in the same list of possible things to do in a regular Action frame (e.g. disengage, help, dash, etc), thus everything that allow you to take an Action allow you to take a multiattack.
I would allow a monk/druid to choose if he is using natural armor (if any) or 10+Dex+Wis for AC. I would also allow him to make unarmed attacks instead his animal routine and use martial arts or flurry of blows.
You gain a bit of temporary hit points but your unarmed damage, ability boosts and monk special abilities (ki, stunning fist, etc.) are delayed by 2 levels... not a big deal.
The main fuss about this, I think, is that certain masters thought it was too op to boost a bear's AC by tinkering on its stats block, thus they were appealing to RAW in order to demonstrate that natural armour is effectively an armour suit, which
is not. It
can't be stacked with unarmoured defence but it surely can't deny it's effectiveness.
The same applies to unarmed attacks. Nothing prevents anyone or anything to portray unarmed attacks, nothing and nobody should prevent a druid monk to discharge a flurry of blows as a bear. Natural weapons are, in fact, not monk weapons, but nothing prevents you to head butt as a bear (or use the... elbows? are those of a bear elbows? o.o') so why should your DM?

if your DM doesn't like it, or he has a very convincing explanation (e.g. grim atmosphere where a "silly" monk bear would not be "cool") or you should call shenanigans on his decision.
"because I say so" it's not a convincing explanation, it's a whine.