Old job doesn't know my employment dates. Application says must be "complete"?

I had a zero-hours part-time job a few years ago that lasted about 2 years.

With only a slight degree of accuracy can I remember the approximate start and end months. I've contacted my old employer and they have no idea where the paperwork is and aren't going to find it.

At the moment I'm discouraged from applying to jobs that make you sign a declaration saying something to the effect of "Your application must be complete, any omissions etc will result in termination of employment or death".

I'm happy to put the approx dates on for jobs that just want a CV and covering letter but for extensive application forms I'd be screwed if they contacted this former employer.

What should I do?

Put the dates in to the best of your knowledge. If you previous employer isn't prepared to find the paperwork for you, they won't be prepared to find it for another employer that asks.

Given that you can't find your payslips, or payments into your bank account and they can't find their 'records' I resume there are none of these and it was a cash in hand job, so no-one is going to be any the wiser than what you put down.