Two path solution.

Build two paths leading from the fence to the enclosed area. They do NOT have to be next to each other.
They can be winding etc, just smooth enough that the Bot doesn't get stuck. I think they would look neat going through your trees and flower beds. Make them out of compacted crushed Granite. Make sure you insert the wire before you compact.
You would program the enclosed area as a separate area so that it would plan on going directly to it.
Install something like this Gate
http://paradiserobotics.com/forum/index.php?topic=12.0 in your fence. You could build it out of wood/chicken wire/whatever. Just make it strong enough so that the kids can't destroy it. If you're worried about other dogs getting in your yard put a second one on the other side. Not too many dogs exist that will craw through 4 feet of tunnel. You could add a flapper that would only flap one way.
Re-layout the wire so that it makes a loop before heading through the tunnel. That will stop it from getting hung up while mowing the Main area. Immediately after the loop insert an island/bush/flower-bed loop that is only one foot wide but runs to the next gate. At that point put a triangle point on it so that the end points towards the returning wire from the enclosed area.
There should be a 3 inch gap so the Bot on returning from the enclosed area does not pick it up.
http://paradiserobotics.com/forum/index.php?topic=11.0Note:
When mowing the Main area it will still go to the enclosed area when it's time to return if it hits the wire that is prior to the first gate.
In the enclosed area loop requires the same layout.
However, if you're using the One Gate layout just drive a stake into the ground to block it from a direct hit that would let it run between the two wires.
Hope that helps toolbelt