Should my Pencil work on my iPad 10th Gen, while iPad is connected to my Apple TV?

The iPad10 fundamentally does not support the second generation Apple Pencil. An iPad10 does support the first generation Pencil - and the new USB-C Pencil.

Here is Apple’s definitive compatibility list for Apple Pencil and iPad:

Apple Pencil compatibility - Apple Support

The first and second generation Pencil use an entirely different pairing and charging interface. 

The second generation Pencil can only be used with iPad models that have the magnetic/inductive dock that is exclusively designed for the Pencil. Charging is controlled via the iPad magnetic dock.

The first generation Apple Pencil uses the physical Lightning port of the Pencil and host iPad for both pairing and charging the first generation Pencil. The first generation Pencil can also be charged using an inline charging adapter and the iPad’s Power Adapter. This pencil can only be used with compatible iPad models that have a Lightning connector; with the sole exception of the new iPad10, requiring a special adapter, iPad models that feature a USB-C port lack any means by which to pair with a first generation Pencil.

In summary, the Apple Pencil models are not interchangeable - and can only be used with the iPad that has appropriate internal hardware.

The use of a compatible Apple Pencil should have absolutely no impact upon (a) your ability to connect to an AppleTV box - or (b) use the Pencil with the host iPad while connected to other devices.