OP as I'm sure you're well aware, there are no easy solutions. But you've asked what you can do, so I'll list them anyway, in case there's something you haven't thought of.
This has been done in a practical way largely with emotion stripped out of it. No, I'm not heartless and yes, I do care - but me caring isn't going to solve anything for you, practical steps might.
You can't afford the cat. It's that simple. It's an awful situation and I'm sorry. Yes, I do realise the cat can't be rehomed and will need to be PTS. But at this moment in time you're choosing between food for yourself or food/litter and vet bills for the cat. If you starve to death, the cat dies anyway. If you end up homeless, the cat has to go too. So you've no option there.
The phone plan needs cutting or the home broadband. You can't afford both. If you've got internet at home you don't need data on your phone, you can use WiFi. If you've got significant data on your phone you don't need home broadband. So one has to go. Cheaper text/calls only phone plans are available. Plenty of people live without home internet. You're going to have to choose.
TV licence has to go. You can't afford it. Get books from the library for free so you're not bored. Or sell the TV and get a portable dvd player second hand and pick them up for 5/£1 from the charity shops as and when you can spare the money.
Whatever you're spending on food and toiletries (you don't list that budget), you're going to have to cut down.
Can you get a bus pass? £50/pcm sounds a lot for local bus use.
The kid needs to be taking cheese sandwiches for lunch and drinking water from the tap from now on. That won't cost £50/pcm. Yeh, it sucks and it'll get boring and maybe they don't like cheese. It's tough luck, the alternative is they don't stay with you at all and you hand over the £10/pcm child maintenance you'd be liable for to their dad, so you can put the £49 saved towards your rent (if the child isn't staying with you there'd be no need of a parenting app either).
There are gaps in your budget, everything you've listed doesn't add up to £1440/pcm. You need to write a comprehensive budget listing every little thing by category, so you can see what to cut.
You need to see what's available to rent cheaper eg a room instead of a flat. If you end up in temporary accommodation through homelessness, that's likely to be only a room anyway and you could be stuck in it for years. So the argument of "needing" a room for DC doesn't fly, you've not got RP status by the sounds of it so DC won't be taken into account by council when housing you, DC has a home with their father. There are no words, I know, but this is where you're at. Denial will get you nowhere, acceptance and practicalities are your only option now. Maybe if you don't have such high utilities or maybe not having to pay CT and if your rent is entirely covered by the UC housing allowance, then if you find a private rental room that'll accept it maybe you can afford to keep the cat 🤷
Whether you decide to look for somewhere cheaper or not, you should still apply to the council. You'll be entitled to a one bedroom place, as a single person without dependants in your household. I don't know what personal possessions you own, but if it's a lot you may as well start selling/decluttering now, because if you get a room in a hostel or B&B for temporary accommodation, you're not going to have room for most of it and you sound like you won't be able to afford a storage unit. Stay put until eviction (not until the eviction date, let them go to court for bailiffs to evict you, else the council won't help and you'll be on the streets).
Don't pay the higher rent if you can't afford it during the eviction process. I think you're supposed to pay the usual amount of rent you've been paying, officially, but I've known people who immediately pay absolutely nothing upon being given eviction notices and nothing much happens to them. Maybe a tanked credit rating, I'm not sure. My advice is to look into potential consequences and weigh up whether being a good citizen is what's best for YOU, in this regard. You can't afford to "be nice" about things at the moment, you need to be ruthless. Your life is going down the toilet, you don't need to feel sorry for or so what's best for, anyone except yourself.
Completely forget about the ex's name on the rental contract. This is not actually a problem for you. It's a potential problem for your ex. If you don't pay rent and they've signed a tenancy, they could be chased for the rent and may legally be liable for it. This is not your problem so ignore it, it's for your ex to sort out themselves. Ditto the new tenancy. If the landlord are in a situation where your tenancy is null and void they'll have a hard time evicting you, so don't worry about it being incorrect. Technically you need a new tenancy because you can't just "take someone's name off", the tenancy has to be ended and a new one issued in a single name. They've just messed it up, that's all. I wouldn't sign the new tenancy agreement, especially if it also specifies the rent increase from the same date (you don't want to be legally liable for it because you can't pay it), but if they force you to then don't worry about the extra name on it, it's their error and they can deal with the consequences of that.
Are you eligible for PIP, if you're incontinent and have anxiety bad enough you're on LCWRA?
Are you claiming all the help you can get for utilities, especially water if you're incontinent? Contact the companies to check.
Are you claiming CT reduction for low income and single adult discount? You should get both.
The knee-jerk reaction to go get a job is an emotional response, not a practical perspective. Assuming you haven't been doing benefit fraud all this time and you've been correctly judged as unfit for work - your dire financial circumstances don't change that.
If you are getting additional help with utilities or extra grants etc, you could lose all that if you lose your benefits, so may be worse off or no better off. I'm not saying people shouldn't work if they can, but if you're genuinely unfit for work and would only be doing it to save being homeless, then there's no point ending up in a worse financial situation than you are now.
You'd also be in a situation where the work is probably going to make you even more unwell than you already are. Then you'd have so much time off sick you'd lose your job anyway.
You'd be in even more debt then, since SSP is way less than UC. Depending on how long you'd worked for the company when you went off sick, you'd be on SSP for upto 6 months, I think, before they can get rid of you for being medically unfit to do the job.
So I'm not sure getting a job is the solution you think it is. You need to properly weigh it up and consider whether it's really realistic.
There's nothing much more I can say except good luck and I hope things improve for you soon 💐