Is this a viable scenario? In other words can a business just forward
the token and have the partner add transaction information for
Generally, no - not because DSS prohibits sharing of a token, but because the token is only useful in the context of the relationship between the processor and the merchant. Random parties P1 - P3 cannot present the token to the processor and have the processor do anything with it; they must themselves have a business relationship with that processor. And the tokens provided by a processor to one merchant may not be portable to other merchants using the same provider.
If not, how does the business that essentially acts as an intermediary
process payment through partners?
It sounds like you're looking for something like the Payment Facilitator, or PayFac*, model. It allows merchant B to act as a middleman between P1/P2/P3 (who are "sub-merchants" in this model) and processor P. The PayFac B will generally work to provide interfaces for sub-merchants to route their payment flow through processor P under the auspices of B, and then will take the aggregated monetary distribution from P and distribute the correct breakdown to P1/P2/P3. The Card data and the tokens will stay with B; customers of P1/P2/P3 will flow through B for future transactions (usually without it being obvious that they're doing so).
"Payment Gateway" is one of the terms you might hear used to describe Payment Facilitators, although I will caution that "Payment Gateway" as a phrase also includes a lot of different non-PayFac setups and means different things to different people.
Essentially, P1/P2/P3 are handing over the complexity of dealing with the cards to the PayFac, B. But they have the benefit of enabling their business by accepting cards.
*PayFac is the Vantiv term for this sort of setup; I don't know if there's a generic term or what the names of competing offerings are. Full disclosure, I am a Vantiv employee. Limiting the example to Vantiv is a sign of my ignorance rather than bias per se :). If anyone names competing equivalents in the comments, I'm happy to bubble them up into the answer itself.