It looks like $A$ and $B$ should be pretty close to being classifiable if one wants to have such an implication. Since the CAR algebra $M_{2^\infty}$ absorb the Jiang-Su algebra $\mathcal{Z}$ tensorically, you always have $A\otimes M_{2^\infty} \cong (A\otimes\mathcal{Z})\otimes M_{2^\infty}$.

In particular, one should assume $\mathcal{Z}$-stability of the C*-algebras in question. But $\mathcal{Z}$-stable, unital, simple, separable, nuclear C*-algebras are conjectured to being classified by the Elliott invariant.

So one maybe has to look for $K$-theoretic obstructions to this implication. By the Künneth formula, one has
$$K_*(A\otimes M_{2^\infty}) = K_*(A)\otimes_{\mathbb{Z}}K_0(M_{2^\infty}) = K_*(A)\otimes_{\mathbb{Z}} \mathbb{Z}\left[\frac{1}{2}\right].$$

One could go in two directions: For one, the implication is trivially true when the C*-algebras $A$ and $B$ absorb $M_{2^\infty}$ tensorially. In $K$-theory, this means by the above formula that the $K$-groups of $A$ and $B$ have to be uniquely $2$-divisible, i.e. multiplying by $2$ yields a group automorphism.

On the other hand, one can ask that $A$ and $B$ be as 'far away' from being $M_{2^\infty}$-stable as possible. On the level of $K$-groups, being very far away from uniquely $2$-divisibility could mean that you have no $x\in K_*(A)$ with $2x=[1_A]_{K_*}$ and also no torsion of powers of $2$ is allowed. I don't know if this is actually enough, but it could be a starting point. I doubt this is a class that can overall be described nicely, though.

Some nice immediate subclass is the set of all unital infinite dimensional UHF-algebras arising as unital subalgebras of
$$ \mathcal{U}_{>2} = \bigotimes_{p\neq2\; \text{prime}} M_{p^\infty} .$$

I hope this helps.