Some agency found me on linkedin and sent me job offer with a fixed rate of £380/day. Upon receiving specifics about the job, I sent my CV and they said they forwarded it to the client. The guy that sent my CV, said the he put me for £360 - that's where I started being suspicious about the rate. He came back to me saying client is interested, but "my manager told me that client is not taking for £380/day". They clearly either lied in the job offer or tried to get as much money for themselves as possible. Either way, my recruitement process stopped at this stage. There was no interview with the client. That was about 3 weeks ago. Now, I found that the client is directly advertising the same job offer (with no specific rate) on linkedin. I didn't sign anything with the agency and they only asked me to send this to them: "I'm happy to be represented by XX via their client YY to the end client at the rate of £380/Day for the position XYZ.". Is it ok for me to approach that client directly now?
The guy that sent my CV, said the he put me for £360 - that's where I started being suspicious about the rate
You should've ran, and ran fast at this point.
He came back to me saying client is interested, but "my manager told me that client is not taking for £380/day".
This is where you ask what are they taking, and if you don't get an answer consisting of 3 numbers, run.
Is it ok for me to approach that client directly now?
Absolutely! Do so and good luck with your application. Maybe explain in the cover letter about the bad recruiter before, as your name may flash some lights in the client's name. I say may because bad recruiters, like the one who approached you, often try to, badly, censor candidate name.
The things that some other suggestions that you may not be allowed to do so is rubbish as You've signed no document to state so. Even if you were it would be extremely unlikely to be binding, as you are waving a lot of right for nothing in return. And in UK uneven contracts like that have generally not been recognized as binding. But you didn't, so even that's not a worry. So go, apply, good luck!
The main question to ask yourself is - Can it do any harm to apply?
My thoughts are, probably not. You have already tweaked your CV and cover letter for this role and the recruiter has already said that it's been turned down and therefore cut ties.
As motosubatsu said, it could cause issues with the agency, but that assumes that the agent was genuine and send in your application. Provided you make sure you mentioned that happened, it's the companies responsibility to ensure they aren't breaking their contract with the agent. To clarify, it's the agent and the company who may have a contract, I'm not suggesting you could get into any deep water here.
Another consideration, is no applying directly for the role, but sending a CV & cover letter, saying:
I've already applied to X through the agent and was told I wasn't a good fit but I am interested in working for the company and would you have any similar positions? If not please do keep me in mind if anything opens up.
The e-mail you sent agreeing to representation by the agency most likely prevents you from applying directly for that position. That's pretty much the point of that sort of thing. Or more accurately it likely prevents the company from hiring you directly.
If the company were to subsequently take you on for the position directly they would be opening themselves up to a potential claim from the agency - assuming that the agency has actually been engaged by the company in some fashion.