There doesn't appear to be any handy hooks/settings in LWP::UserAgent for doing this kind of timing.

Tracking through LWP::UserAgent, there is a fair amount of housekeeping and error checking that has to be done before and after the actual request is sent and the respose received. If your intent is to measure the user experience, then their browser is probably having to do a similar amount of work and so you should count this. However, if your trying to measure network or server response times, you might consider putting your timing points a bit closer to the IO itself.

Before and after the last if else block in LWP::UserAgent, send_request() sub might be a good place or even in the request() sub in LWP::Protocol::HTTP

For doing your timing, I'd recommend Benchmark::Timer. It allows you to have multiple concurrent and overlapping timers, each with its own name. It takes care of storing and averaging the times but still allows you to access the raw data if you need to.