That’s quite a specific criteria- there are numerous event management plug-ins out there with bookings built in, but none will provide exactly what you are after. In general they will all have some form of date-picker (usually jQuery datepicker) – that’s because in general people will want to manually select dates.
My suggestion is to create your own plug-in :D. Or adapt (extend) an existing one. The following plug-ins provide booking registration. In general though, they’ll provide more than you need and not quite the UI you are after – but plug-ins are designed to cater for wide audiences, if you want something more bespoke you’ll have to build it yourself, or pay someone to do it for you.
- http://wordpress.org/extend/plugins/events-manager/
- http://wordpress.org/extend/plugins/events-made-easy/
- http://wordpress.org/extend/plugins/event-registration/
My own plugin Event Organiser, does not provide booking automatically. But this ‘add-on’ does provide an extremely basic interface which you could build on.