Confirmed that using the following commands yields slightly different projects and thus it can be used to “sandbox” using various mix archive-related commands (like phx.new and many others):

MIX_ARCHIVES="$HOME/.mix_phoenix-1.5.5" mix archive.install hex phx_new 1.5.5
mkdir ~/demo155
cd ~/demo155
MIX_ARCHIVES="$HOME/.mix_phoenix-1.5.5" mix phx.new demo

MIX_ARCHIVES="$HOME/.mix_phoenix-1.5.6" mix archive.install hex phx_new 1.5.6
mkdir ~/demo156
cd ~/demo156
MIX_ARCHIVES="$HOME/.mix_phoenix-1.5.6" mix phx.new demo

Doing diff -ur ~/demo155/demo ~/demo156/demo only shows differences between the secret_key_base and signing_salt variables (both irrelevant here) and – voila! – there are differences in mix.exs:

-      {:phoenix, "~> 1.5.5"},
+      {:phoenix, "~> 1.5.6"},

and:

-      {:phoenix_live_dashboard, "~> 0.2"},
+      {:phoenix_live_dashboard, "~> 0.3 or ~> 0.2.9"},

TL;DR: Using the MIX_ARCHIVES env var works perfectly for my goal.

Many thanks to @NobbZ for pointing me in the right direction.