I feel like this should be possible but don't have an hour to sink into messing around with it tonight. The LiveHeaders output (after you configure it to only record pertinent URIs) can be parsed from a text dump. They can then be pretty easily loaded into HTTP::Request objects which you could use, I think, with Mech.

I'm just sort of spitballing it though. I might be leaving something out but I suspect that unless there are transitory checksums or tokens coming from JS that this shouldn't be super hard. I'll come back to it if I have time tomorrow.