The mass number you through out (large portion of the mass of Neptune) seemed really off to me so I looked it up and found some comparisons.

For a flyby: Daedalus mass = 5.0 * 10^7 kg (500 kg of scientific equipment)

With a slowdown to stay in the system: Daedalus mass = 10^10 kg

Mass of Neptune: 10^26 kg

So even for an extended stay, the mass of the Daedalus is no where close to a significant portion of Neptune.

I looked for equivalent mass objects, here is a few:

1.4 * 10^18 = all oceans on earth combined

10^12 kg = 1.4km diameter asteroid (Icarus)

3.5 * 10^10 kg = 0.5 x 0.3 x 0.2 dimension asteroid (Itokawa)

Estimates put the number of asteroids larger than 1km diameter in the belt between mars and Jupiter at 750,000

These are just straight mass conversions, no idea about the real content of fuel. If all those asteroids were straight fuel then you could launch upwards of a million Daedalus class ships that stay in the orbit of their target star.