You paid cash and you have lost your receipt but do you still have the box? That would have the shop barcode label on it proving where you bought them.
The main issue that I see here is the length of time and what caused the straps to come away.
I retail shoes myself and have over the years seen many different faults. It is exceptionally rare for both shoes of a pair to develop the same manufacturing defect in the same places. Left and right shoes are not made at the same time so even though it could be weak leather or faulty stitching it would be a rare case on both shoes at the same time.
Could it be that extra stress from your daughter pulling the straps has caused the problem ? Lots of children do overtighten shoe straps and it's not unfortunately a manufacturing defect.
However, having said that you do still have options. You have had use and wear from the shoes so a full refund is unlikely, you could ask for a repair - it's a fairly cheap job to get done at a cobbler's. Brantano may agree to pay all or some of the cost if you provide a receipt for the repair.
Alternatively contact Hush Puppies themselves, in my experience they are always happy to look at shoes with faults and they will give you an honest report on the shoes. They themselves may exchange them or agree to compensate the repair.
You will need to act fairly quickly though because if your daughter has grown out of the size then they probably won't give you a refund or a replacement. They will know if they are too small from the removable footbed which will show where her toes are in the shoe.
Good luck.
Oh and btw if you do get them repaired expect to pay around £10 - £12 for it.