£100 is v low. It is the equivalent of a £25K salary, if they were paying an employee or less if you take into account them paying on costs. Are your skills/experience worth more than that? I am guessing if you have a day rate of £350 they are....
There is also the issue of tax & the other costs of running your business, which is it difficult to support on £100 a day.
You say they are out of your core 'area' - is their sector/niche somewhere you want to build business? If that is the case there could be more worth trying to make it work. If not, then it sounds like you are getting paid very little for an area you are not really interested in strategically.
As difficult as this is when you are starting out and you don't have much on, you need to think about the opportunity cost. If you do the 8 days at £100 (if I have understood your post) then you earn £800 but you are potentially missing out on £2,800 (8 days at £350). You might kick yourself if you tie yourself up with them and then cannot respond to a better opp.
I think your approach of - "You have £1K, for that you have 3 days or 21 hours" is the only way to go. Definately fix them on the hours/ days, if they are chancers - which I think they might be expecting a freelancer to work for £100 - they could well have very high expectations of what could be delivered for their money.
I have also personally found (marketing freelancer), that people tend to value what you do if you charge a proper rate. I also think £100 to me sounds desperate (sorry if that sounds harsh) and they may well use that to their advantage. I think £350 to me says "I know I am worth it".
You mention the rate not reaching your normal clients. That might be true, but you could end up with a rep amongst this new sector of cheap, so you could find yourself trapped in a low paid cycle etc.
However, this is all very easy for me to say as a stranger. You know how much money you need each month and whether or not you need to take the work.
Good luck.