Hello! First of all, congratulations on getting an interview!
If you have been invited to a first-round interview already, you've cleared a big part of the hurdle in terms of the competitiveness. It might be a bit awkward to ask to change to a different office yourself now, but you could reach out to a recruiter to discuss if you feel strongly about it.
The trade-off between applying to a more desired but also more competitive office vs a less desired and less competitive office is one that many candidates face, and it really comes down to personal preference. Unfortunately, it's hard to objectively assess how likely you would be to get an offer in the NY office versus elsewhere, so you are going to be making this decision based on a subjective assessment of the situation, your risk preference, the degree of your desire to live in NY, and so on. That being said, here are some things I would think about:
1. Your general strength as a candidate - it is impossible to assess this without much more detail about you, but generally speaking, the fact that you managed to get a first round interview from BCG's NY office is a very positive sign. There are going to be many other candidates going through a similar thought process to you, and ultimately some will choose to opt for a less competitive office. On the other hand, BCG is a top-tier firm, and there are going to be many strong candidates recruiting for all of their offices, not just New York. I would take some time to think about how you feel about your candidacy and level of preparation. Of course, this is very difficult to assess by yourself, but you can get an expert opinion by working with a coach, asking a current consultant to help you run a case and get their opinion, or simply case prepping with your friends.
You might also like to know that in some cases where the firm thinks you're a strong candidate but your top choice office is oversubscribed, they may allocate you to your second-choice office further on in the interview process. This happened to me when I was a candidate after my first-round interview, and I recently replied to a question on PrepLounge concerning a similar situation. This is no guarantee of course, but perhaps something to bear in mind.
2. Your risk tolerance/comfort level interviewing at the NY office - something entirely unrelated to your strength as a candidate, how comfortable would you feel interviewing for what you know to be a competitive office? Some people thrive in areas where they feel like they are facing more competition, whereas others will feel more at ease and hence able to perform better in situations they perceive to be less competitive. Both the preference and the degree of competitiveness here are entirely subjective, but I would think about which situation you would feel like you could perform better in.
3. Your second/third choice office - going back to my earlier point, BCG is a top-tier firm, and so every office is going to be quite competitive. In the scenario you outline, what would your alternative option be? Generally, offices in US cities like San Francisco or Boston will be similarly competitive to New York, so if the alternatives you are considering are within that tier, your experience is unlikely to differ substantially.
4. Your other options - are you interviewing with any other consulting firms that you would be excited to work for? Are you interviewing for their New York office? If you are applying to a wider range of companies, maybe you would prefer to only apply to your dream office. If you are putting all your eggs in one basket with BCG, maybe you prefer to be more cautious. It could be helpful to weigh up all your options to assess your best strategy with BCG.
5. Industry / sector interests - what are your industry interests in consulting? Generally, NY offices tend to specialize in PE and finance, so if your industry interests lie there, that could be a natural fit. Alternatively, if you have particular industry interests elsewhere, you could look into which offices work the most with those types of clients.
6. New Jersey office - if you just want the experience of living in NY, you could think about interviewing for the New Jersey office. That way, you could still live in the City!
Ultimately, this is a conundrum that many candidates face, and there are no right answers or guarantees. Think carefully about your options, but at the end of the day it is up to you to decide what is best for you. Best of luck in the process!