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. The 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.
Update March 17th 2018: There are some out there arguing that we should use the Equinoxes to decide what New Moon starts the year. One of these I can't recommend since they're Lunar Sabbath supporters and very Racist.
But the Argument is that Israel in the Wilderness didn't have Barley to check, and neither did Moses on the Ark. And that the Sabbatical and Jubilee years you weren't to be planting any crops. Genesis one says the Sun, Moon and Stars are to be times and seasons are determined.
They also argue that tquwphah Strong Number 8622, means Equinox. It means Circuit but apparently that's kind of what Equonox means. I'm not entirely convinced of that argument however.
However they largely fall into the same issue as the Kariaites, that it should always be the first New Moon after the Spring Equinox, not simply the closest. They say that if you started the year 13 days before the Spring Equnox, most of Tabernacles would happen before the Fall Equinox.
I think the answer isn't closest or first after, I think it's somewhere in between. They're concerned with starting it too soon, but I feel it can be equally a problem to start it too late.
Again, no one is willing to consider that the New Moon that starts the year can maybe be identified as such afterwards.
Take March of 2018 when I'm writing this Update for example. The New Moon (as in the first Crescent) will probably bee seen on the 18th or 19th. In this case the Equinox is certainly soon enough afterwards that it seems extreme to delay the new year a whole month over it.
We need to make sure it's Spring when the Nisan Festivals happen. But the start of Nisan can be sooner.
Update March 22nd 2018:
Another reason I would be more concerned about doing them a Month late then a month early is the Feast of Jeroboam. Jeroboam's Feast wasn't in the 6th month it was in the 8th. The people in Ancient Israel doing Tabernacles a month later where Jeroboamists.
For people attacking Christmas because they vaguely happening at the same time as Pagan holidays is so bad. In a year like this one, not starting Nisan till nearly a full Month after the Equinox is going to place Passover on Walpurgisnacht/Beltaine and Halloween during Tabernacles. But the truth is it's the Kabbalahist holiday Lag BaOmer a month after Passover that has it's origins as a Judaized form of the Mid Spring Pagan bonfire festival.