Video Camera with Detachable Viewfinder and Remote Control
It is probably pretty unlikely in your price range.  Remote monitors are a common feature on high end professional gear where different features of the camera are actually being controlled by multiple operators and reviewed by a director and cinematographer.  Having a removable, touch, control interface however, is an exceedingly rare need in the consumer market.  The closest you will likely come is a wireless, wifi based control that lets you control the camera and record to the camera itself (so minimal lag on the controls) even if there is slight lag in the video feed itself.

You could look for cameras with a firewire interface and remote control software for a tablet.  That would be lower latency than wireless, but I'm not sure that many new cameras include that kind of support.  

You could also consider the more limited option of an HDMI or SDI field monitor and a basic LANC controller or similar.  This would allow you to have a super low latency monitor and super low latency controls, but the controls would be more limited.  It would probably still end up pushing the edge of your price range though.  The monitor itself is generally $100 or more alone and can easily go upwards of $200 for a marginal quality one.