That won't go down well. You either have to do it by the book, as others have suggested, and advise them in advance, or do it covertly. One of the advantages of taking a friend is that you can covertly record the session while your friend gives the appearance of taking copious notes, then transcribe the recording afterwards. But I suspect the OP doesn't need the additional stress of such shenanigans.
As others have said, this is fairly typical of a system that sees advantage in putting the frighteners on people. They don't "already think you're guilty", but they
do like to give you that impression. I suppose, if people really are guilty, it's a quick way to a confession or a slip-up, but to people who are innocent, perhaps naive, and anxious, it makes an already stressful situation ten times worse.
Mell, you'd do well to find out what level of "ex-husband staying over" is considered to be problematic (assuming that information is even available), to reassure yourself that you're in the clear...but the fact that he and you are sharing responsibility for caring for a disabled daughter will, I am sure, help with that. You might want to take some time to write out some notes on what that care involves - whether someone has to get up in the middle of the night, how often, what has to be done, etc. Be prepared to be asked intrusive questions about where he/you sleep (and be prepared to, ah, tilt your answers in the general direction of "on the sofa", etc - you're only countering the assumptions in their filthy suspicious minds).
It sounds like some nosey neighbour has spotted him turning up and apparently staying overnight, put 2 and 2 together, got 5, and then got all malicious. Your neighbour can't prove anything about your sleeping arrangements, and neither can DWP, and you have an utterly reasonable and plausible reason for his being there, although you might take advantage of the interview to ask them if they know if there's any provision for helping you with the cost of paid care for your daughter, while you're at it