You can and should do 90% of your training in light hiking, approach or even running shoes. Rainier in the summer normally does not require double boots. I decent single mountaineering boot will be plenty. I am and many others have climbed it in running shoes with light aluminum strap on crampons. However, you should do some of your final preparation hikes in the boots you will use for the climb. You need to be sure they are comfortable enough and don’t hurt your feet.