Can oxygen gas in air react with Mn3O4 to form MnO2?
Manganese(IV) oxide decomposes around 500 °C (German Wikipedia lists $\pu{450 °C},$ [1, p. 396] lists $\pu{530–585 °C}):$
$$\ce{4 MnO2 -&gt;[\sim\pu{500 °C}] 2 Mn2O3 + O2}\label{rxn:1}\tag{R1}$$
On the other hand, according to [2, p. 12], $\ce{Mn3O4}$ reacts with oxygen only above $\pu{500 °C}$ and forms manganese(III) oxide:
$$\ce{4 Mn3O4 + O2 -&gt;[&gt;\pu{500 °C}] 6 Mn2O3}\label{rxn:2}\tag{R2}$$
Upon further heating, the reaction in the opposite direction takes place [1, p. 396]:
$$\ce{6 Mn2O3 -&gt;[&gt;\pu{940–1090 °C}] 4 Mn3O4 + O2}\label{rxn:3}\tag{R3}$$
All in all, the third reaction you've proposed $(\ce{Mn3O4 + O2 -&gt; 3 MnO2})$ is unlikely to occur and the product will be $\ce{Mn2O3},$ which unfortunately would break the catalytic cycle.
As for the first two reactions, they are described in the literature all right [1, pp. 397, 399]:
$$
\begin{align}
\ce{MnO2 + SO2 &amp;-&gt;[\pu{450 °C}] MnSO4}\tag{1}\\
\ce{3 MnSO4 &amp;-&gt;[\pu{850–1155 °C}] Mn3O4 + SO2 + 2 SO3}\tag{2}\\
\end{align}
$$
Reactions \eqref{rxn:1}–\eqref{rxn:3} are also described in Greenwood's Chemistry of the elements: [3, pp. 1048–1049]:

It is $\ce{MnO2},$ however, which is by far the most important oxide in this group, though it is not the most stable oxide of manganese, decomposing to $\ce{Mn2O3}$ above about $\pu{530 °C}$ and being a useful oxidizing agent. […]
$\ce{Mn3O4}$ is formed when any oxide of manganese is heated to about $\pu{1000 °C}$ in air […].
If $\ce{MnO2}$ is heated less strongly (say $,\pu{800 °C})$ than is required to produce $\ce{Mn3O4},$ then the more stable α-form of $\ce{Mn2O3}$ results […].

References

R. A. Lidin, V. A. Molochko, and L. L. Andreeva, Reactivity of Inorganic Substances, 3rd ed.; Khimia: Moscow, 2000. (in Russian)
Turova, N. Inorganic Chemistry in Tables; Springer: Heidelberg ; New York, 2011. ISBN 978-3-642-20486-9
Greenwood, N. N.; Earnshaw, A. Chemistry of the Elements, 2nd ed.; Butterworth-Heinemann: Oxford; Boston, 1997. ISBN 978-0-7506-3365-9.