I haven’t dealt with the issue but if I had to, I would keep different versions of the program somewhere in directories with names based on the plugin version, e.g. ~/.local/share/elixir-ls/0.15.1.

I would have a symlink called ~/.local/bin/elixir-ls (or somewhere sensible that is in your $PATH) which points to the elixir-ls version in use at any given time, and reference that in the elixir.pathToElixirLS setting in my coc-settings.json file (I use NeoVim, so your config file would be different for e.g. VSCode).

Then I would make a script called elixir-ls-switch-version that looks something like this:

#!/usr/bin/env sh

if [ "$1" = "" ]; then
  echo "You must pass the name of the elixir-ls version you want to use as the first positional parameter."
  exit 2
elif [ "$1" = "ls" ]; then
  echo "Here is a list of installed elixir-ls versions:"
  ls ~/.local/share/elixir-ls
  exit
fi

# make a new symlink containing the desired version of elixir-ls
path_to_new_symlink="~/.local/bin/elixir-ls"

echo "Making a new symlink..."
ln -sf "~/.local/share/elixir-ls/$1/language_server.sh $path_to_new_symlink"

I would place this script in my $PATH (probably in ~/.local/bin/elixir-ls-switch-version) and keep a backup in my handy-dandy scripts git repo.

Then, I would run elixir-ls-switch-version 0.15.1 (or run elixir-ls-switch-version ls if I want to see a list of installed versions).

Then, in NeoVim, I would run :CocRestart (or whatever the equivalent is for your editor of choice) and hope for the best!