The proper way to go about it is detailed by ohwilleke's answer. But for the sake of amusement let's ignore that and consider the fallback:
Petition/Pressure/Obstruct Congress
Reapportionment due to the Census, while required by the constitution, is not automatic and self-executing by default. It requires an act of Congress to determine both how many seats the House of Representatives will have and how many seats each state gets.
As a point of fact, the House was never reapportioned as a result of the 1920 census (a nice summary of the events can be found in Chapter 2 of this law review article from 2018). At that time reapportionment was a political hot potato, with rural states and areas fighting against a loss of representation as urban areas exploded in population. This made reapportionment into a heated battle to decide how it would be done, and in particular decide how many members the House should have. And the urban boon, combined with internal migrations induced by the effects of WWI, meant the population distribution had dramatically altered since 1910. Some areas were now wildly overrepresented and others wildly underrepresented. Disagreements and brazen political obstruction got so bad that the House never came to an agreement. The House apportionment was thus stuck at the what was determined by the Apportionment Act of 1911 for the entire decade.
Eventually they passed the Reapportionment Act of 1929, essentially in force to this day, fixing the number of seats at 435, as well as providing a default method for all future reapportionments. This allowed the House to finally be reapportioned as a result of the 1930 census. While changes have been made, with reapportionment now following the provisions of Title 2 (Section 2) of the USC, the basic thrust has remained the same: fix the size of the House and create a default method and process for future reapportionments.
But the point here being that if you don't want reapportionment to change your state's representation, then a technically valid way of doing so is to try to affect the necessary legislative acts of Congress. A bit of a tall order for a single state, especially with a default process in effect, but nevertheless technically valid.
You could try to get them to pick a different method for reapportionment; the current method is the "method of equal proportions", intended to minimize the disparity in the populations of districts across the US. In a case like the present one with New York, a different method may change the end result so that New York does not lose a representative (though some other state might; New York's retained rep need not be removed from a state that otherwise stood to gain one).
Or you could even try to get them to increase the size of the House of Representatives. The Constitution places a maximum value on this of one representative per 30,000 people, which is currently in the neighborhood of 10,000 I think, so there's plenty of room to work in, constitutionally speaking. The reapportionments of 1911 and earlier had typically been done in such a way as to prevent any state from losing representatives, in fact. This could let a particular state have no net change in its total representatives. It's proportion of representatives in the House would still go down, but it would technically keep the same number of seats.
If you can convince Congress to repeal the current default methods, or have SCOTUS declare it unconstitutional (some have advanced such an argument, but to my knowledge none have advanced it meaningfully through the courts before), then you'd be back to the situation where each reapportionment needs a new act of Congress. If the reapportionment is a partisan issue, a single state's Representatives or Senators could fairly easily take the process hostage to try to either block such legislation entirely (recreating the 1920-1929 lack of reapportionment), or win concessions on the method that manages to favor them. A presidential veto of the apportionment law would also then be on the table, providing Congress wouldn't easily override it (if you failed to get a majority of a chamber to block it, you may still have gotten better than a third on your side).
The issue with any of these ideas, especially in the current context, is that, even devoid of the currently existing default process for reapportionment, the law detailing how reapportionment shall be done was typically enacted before the Census was carried out. In this case, it won't be clear to anyone which states stand to gain/lose/stay-the-same from any given method once the Census data comes in. Of course I'm also intentionally ignoring the practical, political consequences of any of the above acts. There's also the question of whether the courts could step in and dictate or otherwise somehow force a reapportionment, which was (perhaps oddly) never resolved by the 1920 non-appportionment. But this was intended as an amusing hypothetical and technicality, rather than a practical and realistic method.
