How to multiply the length of a pulse?
Could you do this all digitally, using ECL logic?  The SY100E137 is a 8-bit counter that has a maximum clock rate of 1.8 GHz.  The datasheet says " for counters in high-performance ATE time measurement boards.  The counter costs $6.72 at Digi-Key.  You would use two of them for a maximum count of 65535 ns. 

Running at 1 GHz, you would start the counter on the rising edge of the pulse, and sample the incoming pulse train in 1 ns slices.  When the trailing edge is detected, stop the counter, then keep reloading it into a count-down counter 999 times and delay the output until the latter counter goes to zero.