There is no convention that uses a gap followed by a hyphen in a compound.
The construction 'grizzly bear-like animals' would be read as using two premodifiers, 'grizzly' and 'bear-like', as you fear.
One is left with the construction using a single compound premodifier needing two hyphens:
There are many grizzly-bear-like animals.
Mark Nichol explains about the need to tack together the cohesive parts of some premodifiers; though he doesn't give an exactly similar example, just as bear-like needs the tacking together* (even if used as a postmodifier), so do grizzly-bear-like and sabre-toothed-tiger-like.
I'll add a (not very elegant, I admit) variant of a classic example of the need for disambiguation where the premodifiers may possibly be grouped in ways with different meanings:
She gave him one of those sweet shop-girl smiles.
She gave him one of those sweet-shop-girl smiles.
*bearlike is given as a solid compound in most dictionaries, but this doesn't seem to be the favoured web spelling. In any case, I'd certainly opt for grizzly-bear-like over grizzly-bearlike or grizzlybearlike.