I was able to get this working using a gem instead of an Engine. I simply could not make Sinatra use a Rails Engine, but it will use a custom gem just fine. So, I added this to the Rails app and Sinatra app Gemfiles:
gem 'app_core', :path => '../app_core'
and then in the Sinatra app main file I had to do this:
require 'bundler/setup'
require 'app_core'
Voila!