Groundhog,
I wouldnt lower my prices either, and I certainly wouldnt be prepared to lower the standard of work, so my advice would be to ensure that all my paperwork is in order, risk assessments/method statements/insurance etc, and just submit the price, perhaps in person, to explain the huge difference in price, ie. that your costs are higher due to the fact that you are a professional and well aware of all the rules and regs.
Your prospect is looking to get rid of the other cleaner for some reason, might be useful to see if you can find out why, which may give you further ideas on how to sell to this particular prospect.
I understand once he gets your price, you probably wont get the job, but surely it would be better to walk away, than try to compete with the cowboy element or try to deal with a person who insists on low prices, just my thoughts.
Regards,
Rob