After doing some initial research, it was looking as though I would need to update the system from the analytics.js code base to the gtag.js (Google Tag Manager) system. With how integrated the analytics.js system was, this wasn't looking like a task I wanted to undertake without first seeing if analytics.js could trigger the conversion events.
The two accounts are already linked, so that wasn't the problem. I was about ready to give up when I ran across a search that gave the this support article from Google.
It looks like in September of 2017, Google implemented the Google Ad cookie in to the analytics.js code base. I will dig through this article and review the Google documentation on auto-tagging.