I had argued before strongly for the Karaite reckoning of when to begin a Hebrew Year. And I certainly still reject the Rabbinic calendar and the Samaritan one.
Aviv is a name for the month now commonly called Nisan used in The Torah. Because Aviv is a term related to the Barley Harvest (As someone not at all a Harvest expert I still don't quite understand it) the Karaite assumption is the Aviv has to happen before the month can begin.
The issue is in a case like this year where the Karaites and the Rabbis both had a second Adar, and celebrate Passover and Unleavened Bread in the 20s of April rather then the 20s of March. Te Barley Harvest is done before Passover happens meaning it's not possible to offer the First Fruits Wave Sheaf on the Morning after the Sabbath after Passover.
(Side note, I still agree with the Karaite view that Purim should be celebrated in 1st Adar when there are two, Esther defines it as being the 12th Month, but the Rabbis like doing it a month before Passover no matter what.)
I think the Goal should be to make sure the Barley Harvest is going on when Unleavened Bread happens.
What I think many people don't want to accept is that it may be the New Moon that begins the New Year might need to be not recognized as such till after the fact. The Torah does not call for anything to be done on the New Years Day itself that isn't called for for every New Moon, the only New Moon that is a Holy Day is the New Moon of Tishri which marks the Midway Point of the year.
Ezekiel 45 makes the First Day of the First Month important (and the Seventh Day of the First Month also), it's unclear to me if those are for every year or just about when that Temple is first instituted. Whichever case that is, Ezekiel's Temple is instituted either at the start of The Millennium or the New Heaven and New Earth, either way when The Messiah reigns, so calendar ambiguities will no longer be a concern.
I think if the Barley is Aviv very soon after a New Moon, like within a Week, then that New Moon should be retroactively considered the start of the New Year.
Josephus said the first month of the Jewish year was Aries. Back then the Sun was in Aries a month before when it is now, beginning around the Spring Equinox and ending in the midst of April. I would not build doctrine on Josephus, and his goal was to define things in terms a Greek audience would understand, but it's still an interesting reference.
That there is a risk of missing Passover is part of why the Second Passover law was instituted in Numbers 9.