I'm not sure I understand what your problem is and why you want to add the complexity of another bridge table and the joins that go with it.
Having multiple joins to the date dimension is quite common. For instance, consider legal activity. You have date suit filed, date arbitration offered, date plea entered, date trial begins, date trial ends, date of appeal, statute of limitations date, etc.
Can you, perhaps, elaborate on your situation and/or why you don't want to have three joins to the date dimension?
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The reasonable man adapts himself to the world. The unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man. - George Bernard Shaw