This does sound like a very reasonable plan. It depends of course on how fit you are and how much the Shasta climb takes out of you. But it should only be 2 days and so even if it s pretty demanding, a week should allow you enough time to rest up and be ready to put in some training before Rainier. You are wise to plan on a training block between these climbs. You won’t want to push too far to build new fitness and make yourself overly tired. But doing weekly training loads similar to the average of what you have been doing in the month or so leading to Shasta should not be too tough for you. Having a light week prior to Rainier is smart, especially if it involves travel.
Good luck on these climbs.