Alternative to Dr. Bronner's?

I have used and loved Dr. Bronner’s pure castile liquid soap for years. I use it as an all-purpose soap – sinks, shower, face wash, shaving, etc.. Recently, however, I have started to get eczema. I researched online, and found that Dr. Bronner’s has the reputation of being exceedingly harsh.

What would a good replacement be?

My criteria are:

Gentler on the skin than Dr. Bronner’s
As cheap or cheaper – I like to buy a couple gallons of soap and refill for a few years. I do not want to have to have to throw out a bunch of plastic waste. As such, I am pretty price-sensitive.
Available in bulk – nice, but not necessary. If I could buy in one gallon containers it would be great, but it is fine if this is not possible.
I did a bit of preliminary research and found Dr. Woods. Does anyone have experience with that? Or are there other recommendations?

Dr. Bronner's is made with coconut oil. I've read that due to its ability to cleanse well (strip oils) it can irritate some people, even though Dr. B's leaves the glycerin in their soap, unlike many commercial soaps. I'm surprised at your ability to shave with Dr. B's. I tried it once, and was not satisfied.

I have a feeling that your best bet is going to be one cleansing product for household use, and another for personal washing. You could keep using Dr. B's soap, or switch to their Sal Suds for cleaning. Sal comes in bulk and is a synthetic detergent. Then find another product made from an oil that isn't coconut or palm oil.

Dr. Wood's soap is in part made from:
Saponified Coconut Oil (with Retained Glycerin), Saponified Hemp Oil (with Retained Glycerin), Saponified Olea Europa (Olive) Fruit Oil (with Retained Glycerin) . Hemp & Olive oils are more mild, but Coconut Oil is the first ingredient. I'm not sure if it would be better for you.

Your best bet might be to buy soap base from a soap supply shop. As an example, Brambleberry ( https://www.brambleberry.com/liquid-bases-c15.aspx ) will sell you 2lbs of 100% olive oil soap paste (makes 1 gallon) for $18+shipping, or $24.78+shipping if you don't want to go through the trouble of mixing it with water yourself.

Hope this helps!