Bank references are pretty standard. However banks can be slow to provide them. The last letting I did, the agent had to seek six months bank statements instead.
What the landlord really wants to know is whether you have enough income to cover the rent, and a level of assurance that you are responsible and will use your HB to pay the rent.
I mainly let in Central London, so no HB. However I would have thought that the bank reference (or six months statements) becomes doubly important when you can't get an employers reference. Property is expensive and a landlord will want to make sure their's is not at risk.
In terms of insurance, the landlord should be declaring the tenancy, and I assume the insurer may well have terms which require proper referencing (I always go through agents who always take bank references, so I've actually never checked.) This is the landlords business, but it is reasonable for them to know your source of income, and have a bank reference.
I would email or speak to the landlord, offer them copies of your last six months bank statements, and briefly explain your circumstances. Say you really love the property and if possible explain future plans, especially if they are on the lines of when your daughter is nursery age you will be seeking a part time job.
The latter may work well as landlords often prefer people who are going to be out all day, as there is less wear and tear. And to be honest they often prefer people without children and pets for the same reason. (Others feel free to flame but no point not saying it as it is.) Another problem with HB is that it can take time to sort out. For example a while ago a friend had a lovely HB tenant, who then got some short term work. HB was stopped and it took ages for it to be reinstated. No rent during that period.
If your guarantor is able to help, you might also offer a larger than standard deposit, or paying some initial rent in advance. (Common now in Central London. Apparently rich international students now form the dominant group of renters, having over taken bankers. These students will normally pay six months or a year's rent in advance.)
If your prospective landlord is a human being (not all are!) they will appreciate the openness and the reassurance you are offering. Even if they are reluctant to let to HB, they might then see you as a person and so be more willing to help you back on your feet. You sound nice from your post, so assume you are in real life.
If it does not work, you gave it your best shot. Tenancies don't work out for all sorts of reasons. If it does, really look after the property and pay your rent on time. A good previous landlord's reference counts for a lot.