Usually when dealing with recruitment agencies and applying for certain IT job positions, at some point they share the client's name (either by phone or e-mail) and usually saying that I should keep the client's name for myself (confidential). This is especially the case when the interview has been already arranged. For example stating:
Please keep the details of the company strictly confidential.
However when other recruiters from different agencies call me, at the beginning they're trying to be friendly and ask me a standard set of questions about my situation and whether I already applied somewhere or have/had any interviews. When I confirm that I have, they start their psychological investigation by asking further 'where' and with 'whom' unless they find what they need to know.
So at this point, I've got a few choices:
Tell them that the client's name is confidential and I cannot tell them that.
After that they usually trying to get that name even harder by saying they'll keep it confidential as well, or saying that they cannot forward my application unless they're sure it's for a different client, as they want to avoid sending my CV to the role for which I already have been forwarded by the other agency. The longer I keep arguing with them the less pleasant the conversation gets (sharing just the location usually doesn't satisfy them, as they claim to have plenty of companies in such area). They basically say I can't apply for the same role twice, so they're just trying to help me, but I need to tell them the client's name, otherwise they get rude.
Ask the recruiter for their client's name.
I've tried this approach a few times, but usually conversation ends with a rude tone. So when they ask me the client's name in order to avoid being sent twice (first by another agency), I'm asking them to tell their client's name, so I can confirm whether it's the same or not. So they usually tell me that they cannot share their client's name at this point (and at the same time they expect that I can). As usually I have to confirm in writing (e-mail) that I'm happy to be represented for this role by their agency, but they probably cannot send me the representation e-mail request for the role unless they're sure that I'm not represented already by some other agency for that role (so I think it's a bit circular logic). Maybe they can send me, but first I need to know what to tell them on the first call (the topic of this question).
Lying to them about forgetting the client's name, as I've got to check the e-mail.
They usually either make it ironic or want me to check my e-mail and tell them, otherwise they don't want to be in the situation where I'm applying to the same client. Sometimes they say that legally they can't forward my application unless they're sure, which probably is a lie, they simply don't want to waste their effort and time. Sometimes they making it ironic and laugh that for example I don't remember with who I've got the interview. Sometimes they say telling the name helps to let the client know that the candidate has huge demand as per interview with X and Y (especially when the names are recognizeable), so they can stress their client and make them rush with their interview, pushing my role quicker.
Tell them the client's name by breaking confidentiality with the other agency.
It's the easiest choice, but in most of the cases it's not enough for them as after telling the name they usually claim they 'closely' cooperate with them as well and know plenty of people there, they ask me further questions about surnames or they ask whether Mr X is still working there (while searching for some random names in their database for that company, claiming they know them) or with whom I've got the interview, while laughing about how many people they have placed there. Sometimes when I say the client's name too quickly or in a distorted way, they ask me to spell it so they can write it down. The goal of this conversation is most likely to gain the relevant contacts, so they can chase this client for any available roles. If I give them any names, it's easier for them to chase them directly. So I usually share some random names when I know they're no longer working there.
Tell them a lie that I don't have any pending applications or interviews.
This usually is the easiest way, but at the same time they can think of me as a weak candidate (as I'm looking for roles for few weeks and nobody wants me). And I also risk being sent to the same company, which they're trying to avoid to begin with.
My ideal scenario would be to avoid telling the client's name without being rude or getting into another train of 'why' questions and being psychologically attacked. My goal is to not sound rude, and keep the conversation and lies to the minimum.
What would be most effective way of dealing with such a scenario?