An airport hotel at Heathrow is really only recommended if you have a very early flight. Otherwise it is just a dreary box miles fromanything fun that you have to spend £20 a day going to and from.
Any PI or Travelodge is just a dreary box wherever you are, no different at an airport. The extra transport cost is £6.40 per day (not £20), as the Central London daily fares cap is £8.90 per day, the Zone 1 to 6 (Heathrow is in Zone 6) is £16.30. So while it costs £15.50 to go one way to Central London on the Elizabeth Line, a return is only 80p more, and it includes all travel in London that day. The potential cost saving is well worth it. Like many Brits for about 10 years I had a commute of over an hour daily from Essex to Central London on affordability grounds, so the time factor is barely relevant, at least to me.
Any Premier Inn or Travelodge will be fine, they are the go to budget hotel chains in the UK, you can put "Central London" as a search term, although even they are £150-£200 a night now during summer months.
On the Travelodge website they have a 'Best Price Finder' - you can set that to Central London or London Zones 1 and 2' then by increasing price, so can rapidly get an overview of the best prices. I can readily see quite a few which fall within your budget. I'm not sure I would stay in a Travelodge anywhere for 4 or 5 nights, but that is by the by.
You could also look at Ibis (Accor), see what they have. They seem to have some quite good options eg (from a quick look, again sorted by price)- the Ibis Blackfriars for £517 for the 4 nights (£567 with breakfast). I'd happily do Ibis for 4 nights. Give me a Travelodge/Ibis option , and Ibis wins without a contest.
One other very interesting option I have become aware of recently is that Wetherspoons have a London hotel. That is another budget option, but one which is head and shoulders above even Premier Inn. It is the Greenwood Hotel at Northolt. Now I totally accept that Northolt is not a Central London location, but it is a quick bus ride from Northolt Tube station and a probably 10 minute walk from Northolt Park Railway station- trains to Marylebone. Northolt Park is a London rarity in that trains are about half hourly (even hourly at times) but the last back is at 2340. Amazingly at 2342 it even has a call on a train from Stratford-on-Avon. For the 4 nights in question they are on for £273 or £303 currently. Personally (if it was me) that is a very good compromise. Bus 487 stops right outside which links to several other railway or tube stations with normal London frequency, and even Night Bus N118 to Trafalgar Square stops outside.