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NSA is the main bus operator on the route. At Busbud we usually sell their tickets and had figured out some issues around booking for foreigners. Unfortunately, it looks like our connection to their system is down right now. We'll investigate.
Here's a schedule for Asunción to Ciudad del Este from NSA's site. Good news is you would be able to get on a bus at 1a or 1:30a the night you get in, if your flight and immigration are timely.
For the way back, you have to get there the night before. There's a bus leaving at midnight but the trip takes five to six hours and you need time to reach the airport and pass security. Plus it'd be wise to build in a buffer in case of a delay.
If you're trying to avoid Ciudad del Este entirely, flying into Asunción isn't the way to do it since buses to Brazil will transit the city and cross the border there. I'm not aware of any direct buses from Asunción to the Argentinian side.
NSA do have a couple of the buses a day between Asunción and Ciudad del Este that continue across to Foz do Iguaçu. There can be a lot of messiness around border formalities though, like bus drivers not stopping for you to get the proper records, so there are risks to that too. My understanding is you should do your homework and insist on things being done the proper way regardless of how you cross the border.