Consider the following situation: Suppose $X$ is a Banach space such that for each finite metric space $M$ and each $\epsilon > 0$ for which $M$ bi-lipschitz embeds into $X$ with Lipschitz constant $1+\epsilon$, then $M$ isometrically embeds into $X$.

Is there anything known about the properties of such Banach spaces?

It follows from results of Ball that $\ell_p$ has this property for each $p$ (any $n$ point metric space embedding into $\ell_p$ embeds into $\ell_p^{n \choose 2}$, thus a compactness argument gives us that $M$ embeds into $\ell_p$ isometrically by passing to the limit.) Similarly the result is true in Hilbert spaces (much more easily.)

(Balls argument gives the result slightly more generally, but it's not particularly important.)

I am pretty sure the result isn't true in general, or at least, I can see no reason why it should be true: I think the space $(\oplus \ell^2_{p_n})_2$ with $p_n$ tending to 1 from above and the Hamming cube provide a counterexample (with some maybe slight modifications). If this doesn't work, in general there's no reason compactness should be applicable anywhere without really really nice properties (like the $\ell_p$ spaces where we can cut dimensions.)

Edit 1: It might be worth noticing (very, very trivially) that $C[0,1]$ and $\ell_\infty$ have this property, however apart from this I can't think of any other examples

I also can not show that even that if two Banach spaces $X,Y$ have the property that $X \oplus_p Y$ has the property for any $p$. I don't think such a statement should be true in general, but that's a real guess.