Unless you plan on digging up your entire garden and possibly also some of your lawn to a depth of about two feet and manally removing all bellflower roots and shoots, then natural is out. Chemical is dicy too, as bellflower has some resistance to roundup. So either learn to love it, or prepare for a multi-year battle based on roundup. On the plus side, the shoots do pull out easily, so you can manually weed it from areas where you want other plants, but it will be an ongoing battle.

I have no creeping bellflower but my place is totally invaded with Creeping Charlie.Gave up on it a long time ago. Has even moved into the bush at the side of the lawn ( well lawn is really giving it credit). And it is in the pasture as well. The lady who gave me some with out warning me about it is long gone but I hope her new job is pulling up this stuff for eternity. But it does have pretty blue flowers.Not all bad. Better that thistles and plantain.

My whole neighbourhood is overrun with it. I have used Round Up on it and it is moderately effective. It will kill it but sometimes there is weak re-growth from the root next year which must then be dug out. It is quite effective for first year seedlings.

I have managed to control it in my flowerbeds but it's still rampant in the hedges and parts of the lawn. My only advice is never let it go to seed. If you don't have time to dig roots in the non-flowerbed areas, at least walk around at this time of year and uproot the flowering stalks. (for some reason they separate from the roots easily) The seeds remain viable in the ground for years.

If it's important to you, you might consider having your soil removed via bobcat and replaced with clean soil. Then keep on top of it religiously.