I have always gone on record suggesting that whatever version of the name of Hatshepsut you wish to use is fine. You are- after all- talking about the same person, no matter which choice of spelling suits you. The same goes for place names and most things Ancient Egypt.
Although as a general rule i still subscribe to the above approach - i have since been able to test this and have noticed that the approach does have flaws.
Consider the countries that have reports on Ancient Egypt (it may be easier to just list those which do not). I will narrow it down to the name "Hatshepsut" for the purpose of this post.
Now consider that not all reporters are comfortable with the spelling "Hatshepsut". For example, Dr. Karl Leser runs the best Hatshepsut site on the web. German born, Dr Leser is more comfortable using the spelling "Hatschepsut". Further websites outside the UK have other spellings including "Hatshepsowe" (there will also be more variants).
The conclusion would be that in order to get the most varied results, it may be best to consider all versions as important as each other. In order to have a level of consistency in writing - it is advisable to go with one spelling of a name.
Most search engines will try to help you. When it comes to more scholarly materials it is always best to vary spellings. These places tend to want specific information and do not accept spelling variations.
This may or may not help you, but i am finding it occasionally useful myself.