If I use Exeggcute's "Propagation" Ability to retrieve it from the Discard Pile into my hand, then use something like Ultra Ball or Computer Search to discard Exeggcute, would I be able to use "Propagation" again to retrieve it from the Discard Pile?
Yes, whenever Exeggcute is in the Discard Pile, you may use Propagation during your turn as long as its Ability has not been disabled.
Source: Plasma Freeze FAQ (2013-05-16), TPCi Rules Team (2013-05-16)

Relevant cards

Note: All applicable cards may not be listed below.
BW10_102
BW9_4