If we assume water, mass goes to 5% (assuming the rest of the ship has an average density equal to water[!]), so ln(1/0.95) = 0.051.
This helps a bit.
A round-trip to 100 diameters and back at 1G continuous acceleration requires 13.44 G-hours or a delta-v of about 482 km/sec.
So exhaust velocity = 482/0.051 ~ 9460 km/sec, from the rocket equation.
>> Direct conversion of most of the fuel's mass-energy to kinetic energy.
> Would running the water through a fusion reactor core accomplish that?
No. You would produce a plasma of hydrogen and oxygen, which could be ejected at speed - but not at the required level of performance.
Fusion implies conversion of about 0.7% of the fuel's mass to energy.
Maximum exhaust velocity is about 12% of the speed of light (100% efficient, all the fuel fused and ejected out the back of the rocket).