So you want some random hypothetical person on the intarwebz to explain how some random private DBI wrapper (which you haven't show us anything more of than the names of two routines therein) handles placeholders . . .

Exactly what color are the flying unicorns in your world? And maybe you should be asking them because they probably have a better chance of producing an answer.

... Due to this, I can't use ...

I don't see what's keeping you from doing the parameter binding within your putData()/getData() routines, once you have the appropriate statement handle available...  Roughly sketched:

sub putData {
    my $self = shift;
    my $sql  = shift;
    my %param = @_;

    # get the statement handle
    my $sth = $self->dbh()->prepare($sql);

    # do the param binding
    for my $name (keys %param) {
        $sth->bind_param(":$name", $param{$name});
    }

    # execute
    $sth->execute();
}
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You would then call putData() for example like this

$mydb->putData(
        # the SQL
        'BEGIN 
             myApp.myProc(:foo, :bar, :baz); 
        END;',
        
        # the named params
        foo => "whatever",
        bar => $data,
        baz => $obj->method('arg'),
    );
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Add the appropriate error/exception handling, rollback and whatever you need... and you're done, essentially :)

(I've sketched it in OO style (to avoid having to pass around $dbh, etc.), but there's no reason you couldn't do it similarly without OO...)

P.S.: I'm assuming you're still struggling with the same issue addressed in your previous two threads: Call stored procedure in perl and Getting error while invoking a SQL Procedure — which it's always a good idea to link to in order to provide some context for a followup question...