1) You should definitely be using composer for this. The auto generated file is probably one of many / several cached/ auto-created files. That besides, hardcoding the entry is completely the wrong way to do this. You should add the composer require reference to your composer.json file, testing it locally as needed to get it to work. Commit the newly generated composer.lock file and run composer install on your server.
2) If you're getting a 500 error that means your error log should be populated with a far more useful and exact error. Check out your error log to see what exactly the php compilation issue is. A useful thing to do is to run:
tail -f '/path/to/log'
This opens the log as a stream in your console and lets you see the errors occur in real time. You'll be able to see what is thrown when you try to execute the TCPDF code.
3) If necessary, change servers. You're using composer, you're needing to see error logs, you need a server where you can do these things easily. If your provider isn't given you ssh access or letting you run composer.phar commands, you need to move. You're using tools that require that level of server access and the only way you'll ever be efficient is the way they were designed to work.