I think other responders may be thinking $250 not £250, which makes it closer to $315-320. Currently rates are even higher because some NYC hotels are part of a city agreement to house immigrants, whether there are any or not (recent travel bloggers have covered this). Some say that leaves the other hotels to raise rates higher since there are fewer rooms. Will this remain by June? That is uncertain of course.

You do have time to:
1) make refundable reservations at all the hotels you find that seem decent (make a daily habit of looking at the sites where you have found hotels with consistently good reviews. Make refundable reservations (read that fine print--IHG and Hilton have significantly lower rates for flexible rates) as soon as you see a good rate. If a new one pops up three weeks later, great. Book the new one and cancel the old one! I do this with IHG. Take a few minutes each day and you may find a steal of a deal. Book with the hotel directly. If you find a great deal, try to get higher status, and you may even get free breakfast/perks that come along with it.
2) Meanwhile, after you make a list of good hotels, also see how many points it will take and consider if you can accumulate those--because of those resort fees, buying points can pay off here.
3) Ask people you know if they have Hyatt Globalist status with those Guest of Honor passes. We get lots of them--though for June, it will be people who have attained Globalist this year, as current Globalists will have expiration in Feb. 2025. JFK Resorts World (sort of close to JFK airport) gets some good reviews from my friends, including non Hyatt status holders. It is not right in the city. I actually just looked at my Hyatt app. June is already bookable. NYC has a lot of stupid "resort fees," but again, none on points. Sometimes this early prices are extra low and sometimes extra high. But points for some central Manhattan hotels are fixed at 20K-25K points. Some that are a bit farther out but still close enough are in lower Hyatt categories and are 15K points. If you book before points rates are gone or Hyatt changes categories, etc. you are locked in. By contrast, Hilton and Marriott may charge 60K to 100K simply for being located near Times Square or something.
4) One more hotel strategy may be if you have any means of a status match with hotels via your current silver status with the one you do have? Most hotel groups status match. You won't want to do it this soon, but look up a status match site and consider that avenue. I have used it. Also, American Airlines often offers its elites a status match with Hyatt. If you have airlines status you could match to American, you can do that, then cross match. There are ways! This is how I landed in some nice suites at a hotel chain I had no status with while visiting a Baltic country and Scandinavia!