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The recent NY Times article Neptune’s Moon Triton Is Destination of Proposed NASA Mission says (in part):

HOUSTON — Is it time to go back to Neptune?

Scientists representing NASA’s Jet Propulsion Laboratory proposed a spacecraft and mission on Tuesday at the Lunar and Planetary Science Conference in Texas that would explore Triton, Neptune’s largest moon.

Unlike multibillion dollar proposals for spacecraft that the agency has usually sent to the outer solar system, this spacecraft, named Trident, aims to be far less expensive, the mission’s scientists and engineers said, or the price of a small mission to the moon.

I don't know exactly how much "a small mission to the moon." but Beresheet is based on a mission that was about US $25 million^† if I remember correctly, so it could be really amazing if that amount of money can get a spacecraft to the Neptune system and into orbit around Triton.

^†it's US $95 million

I'm not sure it will orbit or if it will be a flyby, but the article says it will image the complete surface and Triton's period is almost six days:

To get to Triton, the spacecraft would fly in a fast, straight trajectory after an orbital assist from Jupiter, similar to the flyby that was used by the New Horizons spacecraft to visit Pluto in 2015. It would rely on a payload of scientific instruments to conduct ocean detection and atmospheric and ionospheric science. The spacecraft would photograph the entirety of Triton, which is the largest object in the solar system that has not yet been fully imaged.

I wrote the article you are referencing. Hobbes, above, has it exactly correct. It is a lightweight vehicle that can launch on even a small rocket, and takes advantage of gravity assists and favorable celestial mechanics to catch Triton at just the right time for encounter with the plumes illuminated. It will image the entirety of Triton in sunlight on approach, and spin around and image it in Neptunelight on the outbound. It basically carries no fuel but for the most modest trajectory corrections, and the hardware is all based on flight tested technologies. It will use radioactive power sources rather than heavy and expensive solar panels (for obvious reasons) and the science instrument payload will allow a fairly robust set of science objectives to be achieved, including ocean detection and characterizing its ionosphere.

Regarding how it affects the Decadal flagship recommendations, the problem facing an ice giants flagship is that you are looking at a minimum two-billion-dollar spacecraft with a 2040 launch at best (flagships take forever for political reasons, and a minimum of 25 years will have elapsed between the launches of Cassini and Clipper), and it would have to launch on something like SLS to get there in a reasonable timeframe (if SLS is still around) at a billion dollars per launch. By then, Trident would have already completed its flyby, and the flagship would have another eight or so years of cruise ahead of it. That's a long time to plan good Triton science based on the Trident science return.

Moreover, I would suggest that if we learned anything from the outcome of the 2013 Decadal, it is that mission sequences might be the way of the future. (See the MSR sequence, and now the Europa Clipper/lander sequence.)

Ultimately, Trident has a long way to go before being selected, and the competition is going to be fierce. (Moon Diver is a particularly thrilling competitor.) But the prospect of expanding the Discovery program from 5AU to 30AU is a real paradigm shift, and would be transformative for outer planets exploration. I hope this helps!

It's a fast flyby in the $500M cost class (a Discovery mission). So not really comparable to Beresheet.

A rare, low Δv trajectory (Fig. 1) enables an MMRTG-powered spacecraft fitting under the Discovery cost cap.

The mission would have to be launched in 2026, for a Neptune encounter in 2038.

New Horizons has effectively demonstrated the scientific value of fast flybys in the outer solar system. Trident’s encounter with Triton will be similarly rapid...

so like New Horizons, they'll have only a few days of close-in data collection. The plan is to do the flyby at an altitude low enough to sample Triton's atmosphere.

During the Jupiter gravity assist, an Io flyby is possible.

More info

There have been more proposals for Uranus and Neptune missions recently, including orbiters and atmospheric entry probes. Interest in the ice giants is increasing, in the current decadal survey they were ranked third after Mars sample return and the Europa mission. Once those are out of the way, it may be possible to get a flagship-class mission funded (which would enable an orbiter at least).
While this flyby is an interesting idea, I suspect it may spoil that process a bit.