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The Neptunian satellite system is unusual, comprising Triton, a large ($\sim2700$ km) moon on a close-in, circular, yet retrograde orbit, flanked by Nereid, the largest irregular satellite ($\sim$300 km) on a highly eccentric orbit. Capture origins have been previously suggested for both moons. Here we explore an alternative in-situ formation model where the two satellites accreted in the circum-Neptunian disk and are imparted irregular and eccentric orbits by a deep planetary encounter with an ice giant (IG), like that predicted in the Nice scenario of early solar system development. We use $N$-body simulations of an IG approaching Neptune to 20 Neptunian radi ($R_\mathrm{Nep}$), through a belt of circular prograde regular satellites at 10-30 $R_\mathrm{Nep}$. We find that half of these primordial satellites remain bound to Neptune and that 0.4-3\% are scattered directly onto wide and eccentric orbits resembling that of Nereid. With better matches to the observed orbit, our model has a success rate comparable to or higher than capture of large Nereid-sized irregular satellites from heliocentric orbit. At the same time, the IG encounter injects a large primordial moon onto a retrograde orbit with specific angular momentum similar to Triton's in 0.3-3\% of our runs. While less efficient than capture scenarios (Agnor & Hamilton 2006), our model does indicate that an in-situ origin for Triton is dynamically possible. We also simulate the post-encounter collisional and tidal orbital evolution of Triton analogue satellites and find they are decoupled from Nereid on timescales of $\sim$$10^4$ years, in agreement with Cuk & Gladman (2005).