Solution uniqueness is an important property for a bargaining model. Rubinstein's (1982) seminal 2-person alternating-offer bargaining game has a unique Subgame Perfect Equilibrium outcome. Is it possible to obtain uniqueness results in the much enlarged setting of multilateral bargaining with a characteristic function? This paper investigates a random-proposer model first studied in Okada (1993) in which each period players have equal probabilities of being selected to make a proposal and bargaining ends after one coalition forms. Focusing on transferable utility environments and Stationary Subgame Perfect Equilibria (SSPE), we find ex ante SSPE payoff uniqueness for symmetric and convex characteristic functions, considerably expanding the conditions under which this model is known to exhibit SSPE payoff uniqueness. Our model includes as a special case a variant of the legislative bargaining model in Baron and Ferejohn (1989), and our results imply (unrestricted) SSPE payoff uniqueness in this case.