A technique I discovered to keep my potato salad from turning to mush is to add vinegar to the cooking water (I also add some sugar and quite a bit of salt to balance it out). The amount of vinegar really depends on how firm one wants the potatoes to remain. One to two tablespoons of cider vinegar per quart of cooking water is the range I use. You might have to experiment because too much and the potatoes will actually be too hard to push through the ricer.
I recently read a Cooks Illustrated article that suggested the same trick and they gave a scientific explanation why it works.
Now, there are a couple of problems for your application. One is the vinegar really flavors the potatoes, which works well for potato salad, but may not work for your application. Two is that I cut the potatoes up into the size I want for the salad before I cook. This wouldn't work for ricing. It remains to be seen if you can get enough surface area on the potatoes for the vinegar to reach and still be big enough to rice well.