Sometimes it's the little things that you just can't live without. (Like your kids, for example.)
My stroller has limited basket space and no storage in the handlebar, either. Enter Kel-Gar's Stroll'r Swivel'rs. These ingenious little hooks hold your purse, shopping bags, child's lunchbox, etc when you have nowhere else to put them. I've read reviews about them breaking easily, but used the same set for years with no problem. Just make sure you put both handles of your bag over both hooks, to distribute the weight evenly. And obviously don't hang anything ridiculously heavy. At $7.92 a pair, even if you have to replace them every once in awhile, it won't break the bank. I keep a second pair in the house just in case.
Note to husbands: Do not use these hooks to hold your dog's leash when you go for a walk. And if you do, for goodness' sake don't tell your wife that's how they broke. She will just give you THE LOOK and walk away, disgusted.