Until we get someone with experience of the hotel lets consider the perspective side;
1. Tripadvisor now has such a reputation for allowing malicious reviews that some hotels are reported to be sueing them
2. even without malicious reporting, many of tripadvisors members don't speak English as a first language and then make inaccurate or misleading posts - a relative of mine recently wasted days trying to book on a recommended train only to to find it hasn't run for 3 years and the website he was directed to doesn't do reservations anyway!!
3. the way these hotel names grow out of control is just asking for the above people to get even more confused "Bestwestern Premier Shaftesbury" is all branding and not unique to any specific hotel. But it is the name of the original Shaftesbury which isn't in Kensington!
4. The Bestwestern website is now so hopelessly complex it just makes mistakes more likely to happen. The two Kensington hotels are shown as seperate entities (not main building and annnexe) but it would be easy to book the wrong one.
So, if you are certain that your booking confirmation from Bestwestern shows Premier I wouldn't worry too much (unless it comes via a third party agent...). But do let us know how you get on.