Luckily, it didn't take much time to figure it out. Thanks to the person who answered here: https://unix.stackexchange.com/questions/16578/resizable-serial-console-window/283206
I was able to use something POSIX standard like the "res()" shell function he described to get it working.
I just put it in as a bash function in my .bashrc file and had it only call when "TERM=linux" which seems reasonable to me for now, since picocom (and the regular Linux pTTY) always seems to set the TERM variable to that.