I've read the related questions, but cannot find an answer for whether the following scenario can be solved on the apple platform.
Is there a good solution for this?
You have several options:
All of those will require one or two taps to install. ;-)
To use TestFlight in iTunes Connect for this, create the iTunes App Store entry as you normally would. Upload the build, and enable beta testing. You'll need the client's email address.
Hockey has a webpage that explains their distribution process: http://hockeyapp.net/features/distribution/
An enterprise certificate would work, but your use case is probably a violation of the terms of service and unless you're a company, you can't get one. I mention it because it technically would work.