If an opponent uses hidden blade on my unit at a battlefield while pushing in with their unit on the same battlefield, and I draw Rengar, Trophy Hunter from the two cards drawn from Hidden Blade, can I play the Rengar to that battlefield?

Yes — you can play Rengar there, just not during Hidden Blade's resolution. You can't act mid-resolution, but once Hidden Blade finishes resolving the chain is empty and you get the next Focus window in the still-ongoing Combat Showdown; killing your unit with an action spell in a combat showdown doesn't end the showdown, both players keep getting Action/Reaction windows until both pass consecutively (FAQ #4223).

Rengar's own permission — "I can [Ambush] to a battlefield where there are enemy units, even if you don't have units there" — is an explicit expansion of the normal location rule, so it doesn't matter that your unit died and you now control nothing there; you can play him as a [Reaction] as long as the enemy unit is still present, and he enters as the Defender (Rule 822.1.b; FAQ #9992/#9820).

The only failure case is if no enemy units remain there when you get priority (an open battlefield), in which case he couldn't be played at all (FAQ #11470).