I'm with Emma. Go with the Bloomsbury flat.
I use AirBnB a lot when traveling and my rule of thumb is at least 15 recent positive reviews, the property owner responds quickly to my inquiries, (where are; the closest tube station, ATM, grocer, coffee shop, etc) and there's a picture of them, not the flat nor their pet nor something iconic like Big Ben in their profile.
The Bloomsbury property ticks all the boxes and is within your price range.
Once in London and you are out and about absorbing the sights, sounds, smells your visit will be authentic.
London is a vibrant multi cultural enclave filled with bold new architecture, iconic buildings, wonderful parks, varied eating establishments, great public transport and, in June, a massive influx of tourists. Pace yourselves. Sit and chat and confirm your absolute must sees then allow flexibility in your planning.
Try to book your Harry Potter Studio Tour in advance to save some money. Picnic in the parks for lunch. Get your you sandwiches, crisps ( the proper term for potato chips) fruit and drinks at a Pret a Manager, or Waitross or Marks and Spencer.
Or enjoy a nice lunch or dinner at a Pub where your teens will be welcomed. Look for sign boards on the sidewalks outside the pubs that signify a lunch deal. Or try eating at museum cafes. All very good spots to dine.
My recommendation for London first timers is visiting the Museum of London. It's free and provides an excellent examination of how London became London. There's a Pret a Manager nearby so maybe a nosh before or after you tour the museum.
Lastly, I will be presumptuous given the statement you'll be visiting the HP Studios that there's interest in the Wizard.
After you tour the Tower of London (it's a definite must visit. Be first in line ) walk from there (about 3 blocks) to the glass covered Victorian Leadenhall Market. It's adjacent (actually butts up against) the Blade Runneresque looking Lloyds of London building. I love the dichotomy of the architecture.
There's actually a glimpse of the market and the Lloyds of London building in the 2017 Jaguar XE currently running in the states. The red car exits the Lloyds building turning right towards the adjacent Market. Movie fans will recognize British actor Mark Strong's voice over for the spot. https://www.ispot.tv/ad/AS2T/2017-jaguar-xe-accolades
While admiring the market place have the teens search for the entrance to the Leaky Cauldron and if true fans they will note that the market is where some of the Diagon Alley scenes were filmed.
Feel free to PM me about fun not so touristy spots to explore while there.
It's a great city. Enjoy your first visit. You'll be back!