How old are the teens?
The Premier Inn has family rooms but they are limited to 2 adults and 2 children under 16. If the teens are older, you’d need an extra room. I’d start by searching on the Premier Inn website to see if they have something that works for you. If not, booking.com lets you search for rooms for 4.
Where is the tour staying? You may want to choose something that is on the same tube line as the tour hotel so you can conveniently get to the hotel the next day. While I agree that the tour hotel would be the most convenient, the RS tour hotels are not always economical. I would though try contacting them directly to see if they have availability and what the price would be, letting them know you will be on a RS tour.
Most hotels will hold your luggage before check-in and after you check-out.
While the hotels at the airport tend to be less expensive, I would not recommend staying there on your arrival day. Going back and forth between the city and the airport will be time consuming so you are better off heading into the city.
Once you have narrowed down your choices, you can come back and post your choices and people can provide opinions on the location.