If Alexander had only wanted to deny any enemy reinforcements from reaching the other side of the island, he would have simply had a section of the bridge destroyed, and then put a small garrison behind.
With just a few good crossbowmen, he was very confident in being able to retard any repair attempts made by the other side, being more than adequate to hold Lord Parker at bay for a very long period of time.
But denying access to just one side of the bridge was not quite enough for Alexander.
He wanted to take control of the whole of it.
Because upon seeing Lord Parker's fleet, nascent as it may be, Alexander quickly came to the realization that the enemy's naval capabilities had not been as squashed as he thought it might have been.