Hendrik W. Lenstra has just informed me that the answer to my question is "no", both in the case of $r = \dim U$ (so we are looking at injective $A$-linear maps) and in the case of $r = \dim V$ (so we are looking at surjective $A$-linear maps). Counterexamples can be found in Exercise 21 (c) and Exercise 22 of his notes The unit theorem for finite-dimensional algebras (arXiv:1703.07273v1).

Thanks, Hendrik, for resolving this nagging question!