Yep for £300 total for 4 you need to rethink (but lunch does give you more options than dinner!). Some already suggested so also seconded. Some you might want to check the menu for prices…
The Wolseley is classic British, likely to see a celeb in there but not especially trendy. Can feel quite ‘brasserie’ but I like it. 4 can have lunch for £300
Rules in Covent Garden is proper old school British, one of the oldest restaurants in London. Think steak and mash, bangers and mash….
Brat, Shoreditch. Welsh Michelin chef’s place. Unassuming building but chef is the dog’s bollocks.
Rochelle Canteen, Shoreditch. Looks quite informal but gets rave reviews.
Dean Street Townhouse, decent. Potential to celebrity spot but will also be busy with members!
The NoMad, Covent Garden. Not cheap but lovely setting in a big greenhouse thing. Good food and cocktails, check it’s within your budget before suggesting though!
Bob Bob Ricard is fun, buttons on the tables to order more drinks (as mentioned above). The lobster mac & cheese is delish.
Temper, bbq / steak place in soho. Buzzy and really good food. Went there with a ‘very particular’ chef friend (he chose it).
Fitz’s Brasserie, Bloomsbury. Steaks! Modern take on British food.
Any of Angela Hartnett’s places (Murano is one of my absolute faves, Cafe Murano the cheaper option but still nice), Italian but not a massively extensive menu iirc.
Sketch is eye-watering expensive, even for lunch. Went there not so long ago and our bill for 2 was over £500 (lunch in the Library). Afternoon tea might be a better bet there.
Check out some of the hotels. The ME has a rooftop bar / restaurant. It’s a great spot and depending when you want to go could be an option.
Sky Garden is also worth checking out. Not eaten there myself but it’s a good spot and has good views.