Bimodal cochlear implant (CI) users combine electrical stimulation from a CI in one ear with acoustic stimulation through either normal hearing or a hearing aid in the opposite ear. While this bimodal stimulation typically improves speech perception, the degree of improvement varies significantly and can sometimes result in interference effects. This variability is associated with the integration of electric and acoustic signals, which can be influenced by several factors, including temporal mismatch between the two sides. In previous work, we utilized cortical auditory evoked potentials (CAEPs) to estimate the temporal mismatch between the CI stimulation (CIS) side and the acoustic stimulation (AS) side, based on differences in N1 latencies when listening with the CIS alone and the AS alone. Building on this approach, the present study estimates individual temporal mismatch at cortical level through N1 latency of CAEPs and investigates the impact of compensating for this mismatch on speech perception. Behavioral and objective measures of speech perception were conducted in bimodal CI users under three bimodal listening conditions: clinical setting, a setting with compensated temporal mismatch between electric and acoustic stimulation and a setting with a large temporal mismatch of 50 ms between electric and acoustic stimulation. The behavioral measure consisted of a speech understanding test. Objective measures included pupillometry, electroencephalography (EEG) based on cortical auditory evoked potentials (CAEPs), EEG based on selective attention decoding including analysis of parietal alpha power. No significant effect of listening condition on behavioral speech understanding performance was observed, even for the condition with a large temporal mismatch of 50 ms. Similarly, pupillometry did not reveal a significant difference across listening conditions, although it was found to be related to behavioral speech understanding. N1P2 amplitude of CAEPs was greatest under the condition with compensated temporal mismatch. The phase-locking value of CAEPs, the temporal response function related to selective attention decoding, and parietal alpha power all showed a significant improvement when applying temporal mismatch compensation, compared to the condition with a substantial 50 ms temporal mismatch. However, these metrics did not exhibit significant effects when compared to the standard clinical setting condition. These findings emphasize that neural metrics are more sensitive than behavioral measures in detecting interaural mismatch effects. A significant enhancement of CAEPs N1P2 amplitude compared to clinical setting was observed. Other neural metrics showed a limited improvement with compensated listening condition, suggesting insufficient compensation solely in temporal domain.