Continually reading an RFID Tag
This is a reader firmware problem.  Most access control readers default to emulating old-fashioned magstripe readers, so they only report once per "insertion" and mask the rest of their reads, as long as the tag is still visible.

Internally, the tag is constantly repeating its ID -- otherwise, the system would be much less reliable, since there would only be one chance to catch the message.

The "right" solution is to find a reader that's configurable for repeating reports, such as the Olimex MOD-RFID125, but I don't know if any meet your price point.

It's strange that the cheaper reader doesn't report a tag on powerup.  It's possible that it has a sloppy RF powerup ramp, so the tag gets into a weird state because it didn't cleanly initialize.  If you have an oscilloscope and a loop of wire, you can try to put it near the reader and tag to try observing the signal.  You're working with a 125 kHz carrier, which should be slow enough for most scopes to capture.