The RF output of the BH1417F is pin 11. On the Belkin Tunecast and Tunecast II transmitters, there is a T or pi attenuator network and a small base load inductor. The antenna is a short length of wire alongside the input pigtail. These things were done to make the output so weak that adjacent transmitters just a few feet apart couldn't interfere with each other and to comply with FCC output power limits. Your pictures are upside down and too unclear to see the attenuator components but they are substantially similar to the ones shown in figure 10 on page 15 of the datasheet.
http://www.crowcroft.net/kitsrus/bh1417f.pdf
You can remove the attenuator components and the base load inductor and install a jumper to the existing antenna but it's easier to remove just the first series resistor connected to pin 11 and attach your antenna line to pin 11 through a coupling capacitor. Use a regular FM whip antenna (28").
The Belkins also run the IC at a lower than optimal voltage when using internal batteries. The IC can operate from 5V. Try using the externally connected power and see what voltage is being applied to pin 8 (Vcc).
Test your range. Don't interfere with reception of licensed transmitters.