You could try and model your pipe using 1D cross sections with lids. It would handle the travel time correctly. RAS can do pressurized pipe flow using cross sections with lids but it can be finicky to get it to be stable. That would also mean having a 1D/2D connection of some sort. Doable but another source of potential stability problems.
If you can compute the culvert hydrograph you could delay it for a few minutes using the advanced rules capability.
For instance, if the pipe is under inlet control you could dump the flow out of system and then use a separate structure with rules to bring it back in with a time delay.
But I’m not sure how you would do that if it is tailwater/outlet influenced.