You shouldn't have as many issues as a single mother with only 2 children, as I have being a single mother of 4. I've had a RE tell me 'no one wants to rent to a single mother with 4 kids'.
I have only tried once to get private rentals since having DD3 & failed. But that was more because there wasn't much on offer at the time. I've been in public housing since & am hoping to buy so I never have this issue come up again.
If I returned to that area I know I'd struggle. It was a struggle to find a long term rental with only 1 child & a partner.