for you Supermum! Thank you for telling it like it is, there are a lot of myths out there and there are good landlords who don't discriminate.
That said, OP, as supportive as I am of your plight, working part time is a luxury. I split up from XDP with a 3 year old and I switched to full time work as soon as I could (I already worked part time). Like others have said, you get 70% of your childcare covered and as long as you work 30 hours a week or more you get the full time work supplement from tax credits too.
I am disabled and it is tough juggling all this, but it can work. DS is now yr5 at school and he is at a school with wraparound care 8am-6pm. I work my backside off at a job where I make the national average wage and get a small top up of housing benefit simply because I live in a very expensive part of the commuter belt. If I was further away I'd not need the housing benefit (which really only offsets my council tax payment).
My family are close by, but I don't rely on them for childcare. They offered to have DS 2 afternoons per week from earlier on this year now they are retired, before that I had no one else, it was childcare or nothing. They are preparing to move to the coast, I don't know whether I'll join them as once DS is in secondary school it will be easier again.
My point is that it might be scary and you think it not possible, but there is childcare out there that you will get money for. And if you are working then morally I found it easier to accept other benefits, which I am slowly reducing and coming off as my career progresses. But you have to make a choice here. If you choose not to work full time then you really can't complain when you are turned down on housing applications for that. Likewise, if you want to keep the cats then you have to accept not everyone will want you. Go on sites like Gumtree, they're often good for landlords who are less discriminatory.