The Echo Input will output audio at 3.5mm audio jack or bluetooth, while the Beam takes input with HDMI-ARC or optical. So you would need a converter, and a switch if you also want the Beam to play TV audio. It's possible, but doesn't really seem like an ideal solution in my opinion.
Although more expensive, the echo link is more the product you're looking for I think. It can take in an audio signal from your TV (optical) and pass that on to the Beam through optical. It also can play alexa multiroom audio with automatic switching between the two, as I understand it.
My personal opinion though, trying to mix Sonos and echo speakers together for whole home audio is a bad idea. If you prefer the cheaper option than go with echos. If you want Sonos quality sound, then go that route. i get that some rooms may fit better with an echo or echo dot, instead of a Sonos One, so you'll be overpaying, but I think you'll regret the extra complication in the long run. Although, if you think echos are great for every room of the house but the TV room with your beam, then maybe an echo link is a great choice for you.
Danny
