Ok I figured out the issue. The Chilkat gem I'm using has OS-specific versions, and the version I unbundled in vendor/gems is for OSx won't work on Heroku/AWS. So even if the path would have worked correctly in Heroku, the gem itself would not.
The problem I face now is because the Chilkat gem is OS-specific, Bundler will generate a Gemfile.lock file specific to the OS on which bundle install was called (OSx in my case). That OSx-specific Gemfile.lock is incompatible with AWS's machines, so Heroku complains that I can't push the code since the Gemfile.lock is inconsistent with what is expected.
This appears to be a known issue with Bundler (How to add Mac-specific gems to bundle on Mac but not on Linux?). I could just avoid committing Gemfile.lock, but we rely on the version dependency management that Gemfile.lock provides. And I don't want to get all the developers on my team to switch to developing on Linux/Unix. Anyway that's a separate issue so I'm closing this for now. Thanks to the people who commented!