What are different ways to 'elongate' an electrical pulse?
I'm sceptical that you'll be able to retain the pulse energy information through excessive amounts of low pass filtering. The more you slow down the pulse, the smaller the amplitude will get, and the lower the SNR of your sampling process will be.

As for how to measure this signal remember that the fastest scopes available actually have very low sample rates (on the order of 40 kHz). The trick is to use a fast sample-and-hold or track-and-hold circuit. 

For a 1 ns aperture, you should be able to make a T/H circuit with just a few dollars worth of parts for a reasonable price. The challenge will be to syncronize the T/H circuit with your laser pulse. Pretty much any ECL logic family still available will have the timing performance needed for this, but the details of how to do it depend on what signals your laser produces to syncronize with.