You could (should) use chlorine to sanitize the water. Here's a couple ideas that might work:
Once you stop adding the chlorine, it will dissipate out of the water in a day or two, or mostly, and be OK for your garden. The "city" water a lot of us use has some chlorine in it, which we use to water our gardens and lawns, so plant life can tolerate a bit. If you can't wait however long that takes, you could transfer your "used" water into another tub, and allow the chlorine to dissipate while you use the first tub. So two tubs, one sanitized for your use, then a second that is off-gassing the chlorine. The chlorine you use in the first tub for sanitizing won't last a week in the second tub, so your once-a-week schedule will work. You could use your little pump to make the transfer. Or just swap yourself. Use one tub for week, then the other the next week, sanitizing only the one you use, the other allow to off-gas. That's an "organic" way to deal with this.
Or, you could try a chemical that neutralizes chlorine, which would be fast, so you'd only need the tub you have. Many municipalities require its use before a pool owner can drain their pool into the street, so that chlorine doesn't enter natural bodies of water downstream (lakes and rivers). You'd just need to study up a bit to find one that is not harmful to plants. Google "neutralize chlorine" and a bunch of info comes up. Or maybe someone else here has some experience with this...
You can buy an inexpensive pool water test kit to monitor the water for either method, which will tell you when the chlorine is gone and the water is OK for your garden.