Immune response to IgA positive bacteria

Depending on the precise details of the experiment, I don't see a particular reason to suppose that being coated with IgA should mean that they'll elicit an IgA response.

I'm assuming that the "coating with IgA" means that the IgA is binding like an antibody, through the variable region binding site on the antibody to some epitope(s) on the bacteria. The way antibody class switching works is that the epitope binding region of the antibody remains the same while the constant regions switch from IgM to IgG or IgA, etc. That means that the ability to bind to the bacteria appeared in a non-IgA antibody, so the ability to bind, per se, is more or less separate from the isotype of the antibody.

The factors that drive a response toward IgA are heavily influenced by location - a response on a mucosal surface is much more likely to drive IgA than one that's systemic (simplistically, because the cytokines and cell types involved are different in the different locations).

An alternate possibility is that the IgA is binding the bacteria through its constant (Fc) region, analogous to Protein A. If the bacteria have evolved an IgA-binding capability in this way, one might guess that they did so because they need it as a defense, and if it's specific to IgA one might guess that it's because they tend to have an IgA-dominant response. But that's an inference, not a strong argument. As a counterargument to that, Protein A does (sometimes) bind to IgA and that's likely more of a side effect than a primary thing.

So I think the answer is "probably not, but you'd need more information to have a good answer."