It's maybe unusual that you are piping the string into plink rather than putting is as a parameter to the plink command? The examples in the documentation suggest you would write:
plink -load profile string
Also, and more significantly perhaps, have you tried using the -batch argument to plink so it doesn't come back with a prompt?
Disclaimer, I have never used plink and I'm just scanning the documentation.
Those were my #1 and #2 tries. Neither worked. The pipe thing was because I saw it online.
I think the -batch option is to prevent plink's output as a program, like maybe a prompt, not the telnet stuff. It changed nothing to the behavior.
Would something like netcat/nc/ncat work for your application? I suppose you do not need any of the telnet protocol features and just need to send a few bytes via TCP to a specific port? In this case, ncat could work.
That's what I also found but it doesn't work. All these software shenanigans assume a background knowledge that ends up having me googling twelve other things. All of a sudden I'm confronted with so many options and new concepts just because I want to send 6 bytes over twisted pair instead of a serial cable.

At this point making a 1000 foot serial cable with RS-422 is simpler.
I'm not sure how you could that in a polite way with a dumb telnet interface, but you could just start it as a separate window and then wait 2 seconds and kill the task.
Something like this.
start "" /b plink.exe .....
timeout /t 2 >nul
taskkill /im plink.exe /f
But it's not a very nice way of doing it
Yes I suspect there is something in the device I'm telnetting to that also needs something set up. I don't know.
I don't understand why the serial port is so easy and obvious and this telnet thing so obtuse and unobvious.