Well Vagus you have the obligatory NHS website afore you, but i will add a few supplementary words.
It is often quite difficult to differentiate between trochanteric bursitis and sciatica and your description of the pain radiating down the back of your thigh, could be typical of both.
Why has my money gone on Bursitis? Listen to the patient and in 80% of cases they will provide the diagnosis and you have done it eloquently.
You describe the pain as bad in the night, but rarely bothers you during the day. That is uncommon with sciatica which is bothersome all the time, but particularly during the day, whereas bursitis is worse at night but eases or may be absent during the day.
Also lying on it at night, making it worse is more typical of bursitis.
Continue the Nsaid of your choice for at least 2 weeks and if there is no improvement, then see your GP.