There have been lots of options with TCAS installations of antennas. However, the 747-4 would have been delivered from BAC with dual directional TCAS antennas, and yes, transponder antennas top and bottom.
The antennas are monitored electronically, so any failure would be flagged. Due to FOD, etc., most of the directional antenna failures are the bottom ones, btw. That's certainly more convenient for the mechanic than replacing the upper one.
Rate of Closure and Range are the primary determinants of TCAS alerts. You had zero rate of closure, so TCAS wasn't worried about a collision, and hence, no need to display a "No Bearing" alert.
I agree that the directional antennas could not resolve the bearing angle, so dropped the target from display. Note, determining target bearing has nothing to do with collision avoidance: it is there just to help the pilot acquire the traffic visually.
GB