Eventually you will be able to style every bit of a multiselect picklist - but it might be not as easy as you think.
Now interacting with the HTML, CSS and JS is not recommended by Salesforce, since they might change the structure at any time in the future without giving you a notice. As a result, your adaptions may break. But if you can live with that caveat and if you are not afraid to get your hands dirty, you can hack it!
To make it look right, you will have to adjust your CSS manually to play along with generated HTML markup. Just using bootstrap is not enough. I do adaptions usually with jquery along with Firebug to figure out the selectors.
I did this once to make the multipicklist component take 100% of the screen width - not only the few pixels Salesforce is providing for it by default. My issue was, that the component first rendered fine on page load, but messed up later after the first click to add or remove items. It crippled up badly. So be carefull, you may run into traps where you don't expect them. I was able to overcome the rendering issues by overriding some Salesforce JS which is invoked in the event handlers. It ended up to be a bigger hack and it took me an hour or two. But the result was good enough.