As you say, I don't see any legitimate reason for a recruiter to have it though, just a new employer when you begin employment?
Just say "sorry I lost it put me on the emergency Tax code"
I Have started jobs with out a p45 and unless you have complex tax status its not a big deal.
Even if they do hassle you, you can easily say that you just asked for the amount you wanted -- or that you were counting income you've made from independent projects that you won't be able to pursue in this demanding new job.
Of course, this does all hinge on your giving a number that they find reasonable... but then, that's a given.
I did at the time, but I have no desire to work with that recruiter again, especially as they lied to me about the nature of one of the interviews I would be attending. The whole process ended up feeling really shady.
Don't put up with these kinds of demands.
Agencies would get in trouble with the Data Protection Act if they asked to see your P45.
And I just recalled you do not have to provide a p45 its just makes seting up PAYE slightly easier
My advice for the UK based folks is simple - go through an agent, defer all salary discussions as something "you promised to sort out with the agent", and then go for the rate you want, and stick with it.
It is hard, but can be done. Remember, most agents will play along with this, as they are often on a percentage fee basis, so it aligns with their sales-weasel interests too!