With the extensive construction of modern highway bridges, incidents of bridge overturning due to vehicle overloading have become increasingly common, prompting a focus on the anti-overturning design of single-column highway bridges. Custom transport vehicles are typically required to travel along the centerline of the bridge at a uniform, low speed. However, in emergency situations where lane closures occur, custom transport vehicles cannot adhere to the centerline as specified. To investigate whether custom transport vehicles can traverse single-column pier bridges with lateral eccentric loads, this study conducted finite element simulations for five typical vehicle loads and 35 different load conditions. The stability against overturning of single-column pier bridges under the influence of lateral eccentric custom transport vehicle loads was analyzed. The results indicate the following: When the eccentric distance of the heavy vehicle exceeds 1m, the structural stability is negatively correlated with the vehicle load and lateral eccentric distance; Custom transport vehicles of types C-I and C-II can safely pass the bridge under all conditions; Custom transport vehicles of types C-III and C-V need to stay within a lateral eccentric distance of 5.32 m when crossing the single-column bridge; Custom transport vehicles of type C-IV must stay within a lateral eccentric distance of 3.97 m when crossing the single-column bridge; For the bridge studied in this paper, the eccentric load reaching characteristic state 1 can be considered as the limit condition for determining whether the bridge has the right of safe passage.