Theorem 42.11 in A. Schrijver, *Combinatorial Optimization: Polyhedra and Efficiency*, 2003 (he references Brualdi, *Common Transversals and Strong Exchange Systems*, 1970):

Any truncation of a strongly base orderable matroid is strongly base orderable again.

Given that, we may assume (w.l.o.g.) that $r(M_1 \vee M_2) = r(M_1) + r(M_2)$ since if not, then we may take an appropriate truncation $M_1'$ of $M_1$ such that $r(M) = r(M_1') + r(M_2)$. And $r(M_1 \vee M_2) = r(M_1) + r(M_2)$ gives us that the bases of $M_1 \vee M_2$ are given by $B_1 \cup B_2$ such that $B_1 \cap B_2 = \emptyset$ (where $B_i$ is a base of $M_i$). And in this case it is easy to see that we get the desired bijection $f: B_1 \cup B_2 \to B_1' \cup B_2'$ from the bijections $B_1 \to B_2$ and $B_1' \to B_2'$.