Ben you need to be really sure to not overwet this carpet, especially in the places it looks most worn. Really stick the ends of your fingers in to feel how much volume of pile is actually there. As long as there's plenty, I'd go carefully with HWE. If not I'd bonnet them, you can still get a really good visual difference, but a very thorough vac with a good upright is essential.
If you've quoted for HWE but switch to bonneting, you don't need to feel like you're over-charging or conning the customer - you're applying the most appropriate measure, all things considered. As long as you put the same amount of effort into the job and they're happy, you've done what you promised
"We're going to rip them up in a few months" is a load of balls to be honest, people say things like this to save face because they're too bloody tight to buy new carpet! You could go back there in 5 years time and you'll still find the same carpets :
The lord of the manor is always like this, he'll buy 50 year old whisky at £500 a bottle but won't pay for any maintenance :
Why would he bother spending good money on cleaning the carpets and then go and rip them up?
Just be careful he's not trying to get a claim out of you.
I'd work only on 25ft of hose at all times to get the best vac performance from your machine. I'd be a lot happier using more airflow to get more moisture out, don't skimp on the drying and put the fans on a section at a time after you've extracted and groomed. Have fun