The ADN8831 eval board app note, AN-695, say it's designed for "medium power TECs (<4A current)".
This part controls temperature by controlling voltage on an H-Bridge. As long as we use transistors don't overheat (based on their R[DS,on] and thermal resistance), why does the ADN8831 care how much current goes through the TEC?
The part doesn't have a hard constraint on the TEC current. With the external FETs and current sense resistor properly chosen, the current can go above 4A.
Update: It works. I only went up to 5A, but it works. I only needed to change the sense resistor and add a higher current inductor.