Is it permissible to celebrate the birth of the Prophet—may Allah bless him and grant him peace—as a way of thanking Allah for his birth, or to perform a specific act of worship with the intention of thanking Allah for victory in battles such as Badr and Al-Ahzab?

The image in question takes the celebration out of the realm of being a recurring Eid. So, if the celebration comes without specific designation or a fixed date, does not recur in a particular manner, and involves no specific act of worship over time, then there is no harm in it. We base this on the words of Allah Almighty: "And remind them of the days of Allah" [Ibrahim: 5]. The verse refers to Allah's blessings upon them, and performing acts of worship with the intention of thanking Allah Almighty for His blessings is, in general, permissible.