How does one prove Avogadro's law from Gay-Lussac's?
His name was Amedeo Avogadro, not Amidio nor Amadeo nor Avegadro nor Avagadro. The full name was Lorenzo Romano Amedeo Carlo Avogadro, Count of Quaregna and Cerreto.
Avogadro's law states that

equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.

More exactly, it also includes single atoms, e.g. for a case of noble gases, due the way how IUPAC defines a molecule.
But remember the Avogadro law implies ideal gases, similarly as the Gay-Lussac's law,Boyle's law and Charles' law. For ideal gases, sizes of molecules play no role. Both the size and volume of ideal gas molecules are considered negligible.
Real gases have deviations from this ideal behavior. See  van der Waals equation as the most used real gas state equation.

$$\left(p+a \frac{n^2}{V^2}\right)(V-n b)=nRT$$
$$\left(p+ \frac{a}{V_\mathrm{m}^2}\right)(V_\mathrm{m} - b)=RT$$

where parameters $a$ (addressing cohesive forces ) and $b$ (addresing non zero molecular volume) are specific for the particular gas.
Then the molar amount of a real gas for given $V$, $p$ and $T$ is the solution of the cubic equation for  variable $n$ :
$$pV - n ( pb + RT ) + a \frac{n^2}{V} - ab \frac{n^3}{V^2}  = 0 $$
It clearly shows that the same volumes of different real gases, at the same temperature and pressure, contain ( usually slightly ) different counts of molecules/atoms.