In the multi-point incremental launching construction process, without setting temporary piers at the mid-span, it is difficult to ensure the synchronization of the launching equipment at the same pier, which can easily cause the beam to deviate from the design axis, leading to changes in the stress of the beam. Calculating the local maximum stress of the beam when lateral deviation occurs is beneficial for establishing a reasonable and reliable threshold for controlling beam deviations. To address this issue, this paper utilizes a finite element model of the bridge under typical adverse conditions, simulating the lateral deviation and asynchronous jacking of the beam to explore stress variation patterns. The results show that the beam stress increases as the lateral deviation and jacking differences increase, severely affecting the structural safety. To overcome the inefficiency of the finite element modeling method when dealing with multiple working conditions and to establish a reasonable deviation control threshold, this study proposes a beam local stress prediction model under the combined influence of lateral deviation and jacking height differences based on the SSA-SVR algorithm. The model calculates the local maximum stress of the beam for 2500 sets of parameters. An engineering case analysis shows that the prediction model has high calculation accuracy, reliable results, fast computational speed, and high efficiency. Based on the stress prediction results, it is recommended to set the jacking height difference control threshold at 20 mm. When the jacking height difference is less than 20 mm, the local maximum stress value under the combined influence of both parameters should be less than the standard limit. The corresponding lateral deviation control threshold can be dynamically adjusted according to the actual travel difference recorded by the jacking equipment, with the control precision improved to 1 mm.