How can you figure out what encoded signal my IR emitter should put out to register a hit on a laser tag gun?
It may be that you're not looking closely enough at the pulses.  Unfortunately, there's no common standard for laser tag guns - every manufacturer seems to adopt their own data standards.

A simple system may just 100% modulate the 38kHz carrier with a lower frequency signal.  I know of one system that uses a carrier of approximately 36kHz, 100% modulated at 1.8kHz.  Without that extra signal, it's assumed to be background noise and ignored by the receiver.

A more complicated system may be transmitting actual data, perhaps with pulse width modulation of the carrier, or of the lower frequency.  You might be able to deduce that, or just copy it, if you can analyse the signal well enough with a scope.