You've had lots of advice and have been told things you clearly didn't know.
It sounds as though proper recycling would remove this whole problem. You're not entirely clear about what you are recycling - you say it's a lot, but then you cite unflattened boxes as part of the problem. If cardboard is going in the bin, you're nowhere near recycling properly.
Food waste into the food caddy would free up black bin space and make your rubbish less attractive to animals.
Cardboard boxes flattened and put into whatever container you have for recycling will remove a lot of bulk from your waste. Use a knife to score larger boxes so they can be folded small.
Find out exactly what can be recycled. In our area it's the food waste, cardboard, paper, glass, metal and foil, all plastic except black plastic, batteries, fabric, small electronics and various other bits and pieces, plus paid-for garden waste.
Find out if you can get extra recycling containers if those are also overflowing.
Save postage bags and envelopes for reuse.
Save shoe boxes etc for storage/use in school projects.
Once your general waste is reduced, get a bin lock if animals are still a problem.
Make sure the kids are putting things in the right recycling container. Mine will dump leftover food in the bin if they're not reminded.
Please don't just buy another bin and carry on sending vast amounts of recyclable waste to landfill. There's a good reason for councils to restrict bins and it's not just to make life difficult.