You will find a lot of double registered AMHR / AMHA horses at the Nationals (AMHR) and the World (AMHA) but the points collected there go with their AMHR records (at the Nationals) or their AMHA records (at the World). They only place I know of where you could compete and AMHA only registered horse against an AMHR only registered horse would be an open show, where you'd have plenty of Quarter Horses, Paints, Arabs, Grades, etc...
Congress (the Shetland Nationals), the AMHR Nationals, and the AMHA World Show are all held at different locations and dates.
You will find various local shows around the country that combine two or more registries.
For example, last month in Oregon, there was a show that was spread out over four days at one location. The first two days was specifically for AMHR registered horses and awarded AMHR points. The last two days was for AMHA horses and awarded AMHA points.
What was interesting about the above mentioned show was that there were quite a few horses that were doubled registered and the owners stayed for the entire four days and showed their horses in what was technically TWO different shows running back to back.