Q: Cannot connect to Apple TV at client site.

Is the Apple TV on the guest WiFi? Probably, since the phone can AirPlay. I assume you are trying to use AirPlay.

Have you ever successfully used AirPlay from the laptop to an Apple TV; to this Apple TV?

Here is the Apple help on this subject: Use AirPlay to display video from your Mac on an HDTV - Apple Support.

But there sre several threads posted by users who absolutely cannot make AirPlay work from various Mac computers. There seems to be something in recent updates that blocks the process.

If it's an ATV 4, you could try peer-to-peer AirPlay. Turn on bluetooth on the computer, open the AirPlay menu and wait a bit for it to discover the ATV, then try to connect. That bypasses the network completely, and might just work.

To get more help you need to post a lot more details: computer model and software, ATV model and software, what icons you have on your computer screen, what happens when you click on them, what messages you see or do not see, etc.

Guest networks can be quite restrictive, may I suggest you use a cable rather than an Apple TV.

I use wifi for presentations at my own offices, but wouldn't use it on someone else's network unless it was well tried and trusted.