I would honor the concerns about sharing one coach to support the success of the project and support the buy-in of everyone, since it's extremely important to the process that people are comfortable working with their coach. - If everyone had agreed on a single coach, depending on the coach, in my opinion it could work, and support group dynamics.
As others have suggested, consider working with a firm that could provide more than one coach to select; as well as process and practices to support developing the culture that will achieve your organizations' goals and proactively identify what might get in the way.
I'd recommend some group work as well - to articulate shared values; what the shared values look like in practice; to agree on a generative way to address values not being honored; to develop and agree on group process for resolving conflicts, and raising challenging issues. (Which may be present now, given the concern about sharing one coach.) As Roza has noted, and in my experience, working with a firm that can provide two consultants might be very effective for this firm. Two consultants leading group work like this can be very effective on picking up on issues, energy, and in working with personality, style and diversity issues.
All the best to you!