Institutional rules provide natural deadlines for negotiations in legislative bargaining. In the continuous-time bargaining model framework of Ambrus and Lu (2010) we show that as the time horizon of the bargaining increases, equilibrium payoffs with deadline converge to stationary equilibrium payoffs of the infinite-horizon bargaining game. We provide a characterization of these limit payoffs, and show that under a K-majority rule, the payoffs of the K legislators with the lowest relative recognition probabilities have to be equal to each other. Hence, by varying recognition probabilities, possible limit equilibrium payoffs are constrained to a lower-dimensional subset of the set of all possible allocations. This contrasts with the result of Kalandrakis (2006) that in the infinite-horizon Baron and Ferejohn (1989) framework, for any discount factor, any division of the surplus can be achieved as a stationary equilibrium payoff through some choice of recognition probabilities.