Whatever you decide to take, try it out at home first. Personally I wouldn't use Dr Bronner's or any other soap based bar on my hair.
I use a shampoo bar that's not soap based. Much better for my hair. Lots of them out there like Ethique which was mentioned and many bars available on Etsy. One of the members here,
@JillGat makes very nice shampoo and conditioner bars.
For laundry, I actually use a separate product, but the weight is negligible - about 2 ounces. What I use for laundry are these
Breezeo laundry detergent strips. One strip is supposed to be enough for a full load of laundry, so I use half a strip for my daily hand washing. I wash my clothes in a 12 liter dry bag that doubles as my stuff sack for my sleeping gear. As I get into the shower I put my clothes, half the detergent strip into the bag. While I'm waiting for the water to heat up for my shower is use I fill the dry bag, then seal it up and give it a couple of shakes before setting it on the floor. While I'm showering, my clothes are soaking. After I'm dressed I shake the bag a bit more to agitate it, then take the whole thing over to the laundry sinks to rinse. The detergent sheets don't create a lot of suds, so it's easy to rinse out.
Because of the long soak time, my clothes get cleaner with this method than just washing them in the sink or a laundry tub.