We have used something similar to your “another way” for a few years now with complete success. Because not all of our employees have access to the business Intranet, we provide two ways for some mandatory training. One method is we mail disks to their home with an affirmation sheet that they can fill out and return in a self addressed and post metered envelope. We also include in this mailing a link to a simple online one page affirmation that they can log into our LMS and complete online if they prefer (most do because they will get compensated faster). This is what I think you are looking to accomplish.
We also have the entire course online for those who want to complete it that way. Both online methods record compliance in the LMS.
For the returned hard copies, we wrote a small program that allows one of our administrative assistants to quickly enter the pertinent information that gets added to the LMS database same as if it were completed online (we do include a flag, however, that allows us to go back and tell which method was used).
The online affirmation was worded by our legal department and basically says that they have completed the training, have no questions and will comply with contents. To accomplish this you only need a single page with a checkbox and some actions. Add a button (exit works great) that when clicked will first check to see if the checkbox is empty, if it is not (meaning it is checked) then send complete to lms and exit. We also added a warning if exit is pressed with the checkbox empty; it asks them if they are sure they want to exit without completing the training. If they answer yes, the lesson closes without marking their compliance.