I've found a mix seems to work (Vue 2.6.10)

<button type="button" class="btn btn-default" @click="buttonClicked('Back to Dash');[handlerClose(),$router.push({name:'dash',params:{ id:id, previous:componentInfo}})]" role="button">Back to Dash {{ dashNumber }}

buttonClicked() logs the argument to console. So semi-colon separated followed by array of functions seems to work. previousComponentInfo is just object that contains name and template name of the previous component.