Hello
I was in a similar position this time last year, minus the part time working bit. I still managed to get a property. I had some help from my local housing choice team, who advised me of loopholes in the "no DSS" thing that I could try. Apparently, having a job, even if just a few hours a week, is very very helpful. As the landlord will ask if you are working, and you can say truthfully, "yes, I am". They never need to know you are claiming LHA because, as you have pointed out, it goes directly to you.
The only snag is, it's a bit of a gamble in the sense that you won't know how much LHA you are definitely getting until you move in and start paying rent. If you LL is aware you will be claiming, and for example you couldn't afford the property without knowing exactly how much LHA you will be getting, then a pre tenancy determination is useful. This looks at the property in question, BEFORE you agree to rent it, and lets you know how much LHA they would pay towards it. Nice for peace of mind; and you do need the LL to agree to that, so they would be aware then you were planning on claiming LHA. However, a pre tenancy check isn't necessary, and you are perfectly within your rights to just go and view properties, say you are working and rent one. I don't see the problem with that.
My case as I say was a little different, I was a SAHM, so I was upfront with my letting agent and they found me somewhere that accepted DSS. I even managed to take my two cats, for a £100 increase in the deposit. Different letting agents have different attitudes to people claiming LHA. They will all tell you it depends on their individual landlords, who accepts DSS and who won't. But that is bollocks. The letting agents have vastly differing attitudes to benefits, some treated me like dirt, but I just shrugged and shopped around until I found one that was pleasant, respectful and helpful. Remember, they really want you to rent through them, as they get money that way. They can recommend to their LL's that they should consider DSS/LHA tenants, especially if the LL is very keen to rent their property for whatever reason. And LL's are having to be increasingly competitive as they housing market means they will rent out rather than sell - and there are a LOT of people doing that. That means increased choice for you, the tenant, and it means that LL's have to be more understanding of things like DSS anyway, in order to exist in what is now a competitive market. Basically, it's a tenants market
I hope some of that was helpful, it might be worth having a look at your local councils website to see if there is help available for people who are looking to rent privately, you might be able to get some support and advice and that is invaluable. Especially if you say there are zero council properties going. It's the same in my area, and as a result they are really pushing private lets, they will sometimes liaise with the LL on your behalf, and they sent me a long list of all local letting agents who accept kids/DSS.
Good luck, I'm sure you will do just brilliantly