Since the client will not be disconnected by Connect (per @slopp's comment), it seems to me like your best bet will be to throw something in the front-end of the application that triggers refreshes after a given period of time. I.e. you could make the app hard-refresh every hour, every 6 hours, etc.
Ultimately, it is unfortunate that this does not give you guarantees of when / how soon after an update the app refreshes, but it does give you the automatic refreshes (on a configurable timeline) without manual intervention.
WARNING: untested The below JavaScript, for example, forces the page to reload every 4 hours (1000 milliseconds * 60 seconds * 60 minutes * 4 hours
)
function reload_page() {
window.location.reload();
setTimeout(reload_page, 14400000);
}
setTimeout(reload_page, 14400000);
There may be a way to do this from R as well 
I suspect there is a way to make this more "savvy" and "detect" changes to the active bundle on Connect, but I suspect the added complexity / fragility is not worth the effort. Writing JavaScript can be painful 