A room in a flat share for a couple, in central Manchester that including bills for £350? No tenancy? I’d honestly be concerned it could be a scam, or there would be a catch somewhere.
You are also underestimating the hidden costs of moving. Even if you do get your full deposit back from your current place, some of it could be eaten up again in extra costs.
The deposit on your next permanent flat could even be higher again, so you could end up in the red.
You may need to buy some new things/ storage, even if the new place is ‘furnished’. E.g lamps and storage.
Kitchen might not be effectively kitted out/ people may not do their dishes. You may need to buy kitchen equipment if you don’t already have it.
What will you do with your existing belongings? Have you factored in the cost of storage?
Also, next summer you may find yourself in the same position if you don’t have permanent employment ….
Would you not consider looking for a cheaper flat/ permanent room to live in longer term? Save over a longer period?
Moving somewhere for a month or two is a lot to go through for the sake of saving a small amount, when you could live frugally to get yourself though the next few months instead.