Isomorphism classes of special fibers of regular minimal models of elliptic curves

In the LMFDB, the elliptic curves $99.d2: y^2=x^3-1488x+40016$ and $99.d3: y^2=x^3-48x-304$ have the same Kodaira type $I_0^*$ at $p=3$ and the same Tamagawa number. Are the special fibers of their regular minimal models (not their Néron models) at $p=3$ isomorphic as $\mathbb{F}_3$-schemes?

Having the same Kodaira type and the same Tamagawa number does not in general imply that the special fibers are isomorphic. Thus, in the particular case of $99.d2$ and $99.d3$, is the above statement true, and if so, how can one prove it?

More generally, for an $I_0^*$-fiber over a finite field $k$ of odd characteristic, what data, beyond the Kodaira type and the Tamagawa number, are required to determine its $k$-isomorphism class as a $k$-scheme?

Any reference would also be appreciated.