Not possible - I haven't used the BBC Sport app properly, but hazard a guess it uses the device's default player (which is an Android player) and thus this is the only desirable time you want full-screen mode.
Android relies on those buttons - otherwise, how can you exit the app? It's like the home button on iOS - you need it to operate the device.
There are event listeners you can use for the back button and menu button which you can override the default behaviour for:
document.addEventListener('backbutton', function () {
// do something when back is pressed
}, false);
document.addEventListener('menubutton', function () {
// do something when menu is pressed
}, false);
That's about it though. Think in terms of the Samsung Galaxy phones - these are hard buttons and so you can't 'hide' them like you can on stock Android.