If you Knock Out a Pokemon using one of Tyranitar GX's attacks, do you get to take a prize card if the Pokemon is sent to the Lost Zone due to Tyranitar GX's "Lost Out" Ability?
Yes, you do. It is still Knocked Out.
Source: Lost Thunder FAQ (2018-10-20), TPCi Rules Team (2018-11-01)

Relevant cards

Note: All applicable cards may not be listed below.
SM8_121
SM8_203
SM8_223