Hi Brian,
I know it is a bit late, but I thought I'd post in case anyone else is looking for an answer. We had the same issue. The way we got around it was to use another Android device (in this case, my phone) and load ArcGIS Collector and Apk Share onto it. Then we just opened Apk Share and backed up / shared the Collector Apk. Then you can just copy the backed up Collector Apk onto your Trimble device and install it.
Hope that helps others.
Matt