RF proximity sensor around a perimeter
You can use simple magnetic induction from a perimeter wire to a portable coil. If you used an audio frequency such as 10kHz to drive a current thru the perimeter wire, maybe 100mA RMS, this should easily be picked up on a portable but also small multi turn coil. The level may only be a few millivolts at 1m from the perimeter but this can easily be amplified and filtered and converted to a dc level that approximates to received signal level. Because the orientation of the receive coil can cause a null in the signal you might want to use two receive coils at right angles to each other to get full coverage at any angle.

I would, on the receiver want to try and filter out unwanted frequencies such as power AC so a bandpass filter set at the operating frequency would be ideal. This can be built around an op-amp and the op-amp can provide gain although it's probably best not to try and use the same op-amp for filtering and gain. Use a rail-to-rail op amp to make maximum use of possibly a limited battery supply.