How many overwintering onions do you want to grow? How many do you need before the spring grown ones will be ready? Depending on where you are, Japanese onions (overwintering) are not always a success.
Heavy clay soil is one of the few good reasons to grow crops in raised beds but it can be expensive if you do it the wrong way. It seems that if you concentrate on converting one bed with intense work then you can make clay soil useable. I grew up on a clay soil farm which grew crops very successfully, with sometimes the odd soggy corner to annoy my Dad, because we added loads and loads of well rotted farmyard manure. We also ploughed in the autumn and left the furrows open to the frost to break down. We never worked the soil when it was wet.
You can easily use those tactics on your ground to get useable beds fairly quickly. It is simply not worth investing a thousand pounds in adding new soil. You would have to improve the drainage underneath it anyway or you would have effectively the same problem. You have plenty of FYM so as long as it's stood for at least a year you can turn in masses of that to make your onion bed. However, I think the onions would prefer that you had done that last year, as they might not like too much fresh fertility. Onions are usually planted in soil that was heavily manured in the previous year.
Were it me, I would dig in a tonne of sharp grit to the clay under the bed and added local soil perhaps from the path around the bed. Then make my onion bed be mixing in the manure with the gritty clay and cover it, perhaps with wetted cardboard and stra, for the earthworms to work over the winter, then plant spring growing onion sets, or plants you have grown from seed, in spring. I would also get the work done for any other veg beds you want for the summer, now.
Clay soil is really not something you can fix quickly forever.