Don't get hung up on "is the beach any better on the other island" syndrome, stay on one island and visit the other for a day trip. - ferry will run you about just under $12US PP each way and take 45 minutes.

Timothy Beach( on Frigate Bay Caribbean Side Beach) and Royal St. Kitts (across the street from Frigate Bay Atlantic Beach and 12 minute walk from Caribbean side beach) should be in your price range on St. Kits. On Nevis, Oualie Beach (as name implies a beachfront property) or Mount Nevis Hotel (mountain side location - 5 minute drive from beach) should be within your budget.