A softener should not introduce a smell to the house but as with any water treatment device, bacterial control is of extreme importance. You added bleach to the brine tank but running it through a regeneration may not allow enouh contact time to truly sanitize the resin. Chlorine will damage resin over time so regularly sanitizing the resin is not recommended but resin is cheap, so it is not as big of a deal as many people make it out to be. you can also replace the resin in the future with a chlorine resistant resin that will last through a lot more sanitization cycles.
I would recommend sanitizing all of the plumbing by introducing a 50-100 PPM solution of chlorine into the plumbing. Be sure the softener is bypassed at this high chlorine level. Let it sit for 1 hour then flush. At the same time, add 3-5 ounces of bleach to the brine tank and start the regeneration. Once you smell bleach at the drain, shut the softener down and allow the bleach to work for 1 hour. All protocols for sanitizing with bleach call for a 1 hour contact time to fully sanitize. A standard softener may run the salt and bleach solution through the resin in as little as 10-15 minutes depending on the injector size.
Once the softener and plumbing are fully and properly sanitized, see if this lasts a lot longer.
All water systems will have bacteria, even minutes after sanitization. It is a matter of control.
Controlling bacterial formation may solve your problem. An ounce of bleach in the brine tank every couple of months may work fine. This will damage the resin over time, but depending on the quality of the resin, you should get several years of life out of it. Consider a municipal supply is always feeding approximately 1 PPM of chlorine to the softener, and the chlorine level exiting the softener will be reduced by 20-30%, that chlorine is being consumed in the softener. A well supply has no chlorine, so occassionally sanitizing the system is not a big deal.