... 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();
}
[download]

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'),
    );
[download]

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...