WI: The British Assault Dorchester Heights?

After Washington fortified Dorchester Heights, the British considered attacking them to dislodge his army. However, they were dissuaded by a storm, and eventually withdrew. What if the storm hadn't happened, and the British had contested the Heights?
 
Could be a repeat of Bunker (errrr Breed's) Hill but without the pyhric victory.

Although, if they should win overwhelmingly maybe they kill the revolution in its cradle.
 
Most likely a battle very reminiscent of the Battle of Bunker Hill occurs. However unlike the earlier battle, in this battle the British would be outnumbered by the Americans and would likely fail to take the heights and suffer immense casualties as a result. This could well spell an end for William Howe's career.

Depending on how many troops the British throw against the American fortifications, Washington's attack from Charleston may end up defeating the British and taking Boston. Such a decisive defeat would be a huge boon to the Patriot cause. However that would require a lot of things to go Washington's way.

A more likely course of events is that both attacks fail and that the end result of the Siege is similar to OTL save for the casualties taken on both sides.
 
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