I have cut tree branches in the bottom of one raised bed (Hugelkultur), and that's the bed where my summer squash all perished by fungus. So I'm thinking I should remove the wood, and just add some perlite to help the soil retain moisture instead. We are in a high fire danger zone, so any burning of green waste is illegal: I'm not making any biochar in the foreseeable future.
I don’t think the Hugelkultur has anything to do with the squash fungus. There are three general types fungi at work - pathogens (feeding on your squash, if that’s actually the culprit), saprophytes (feeding on detritus, such as the woody material you buried in the ground), and mycorrhizae (the fungus-root symbionts).
A friend of mine with a lot of gardening experience tried Hugelkultur a few years ago. My impression was that the main benefit was supposed to be moisture retention, but the concept seemed pretty strange to me - one reason being there was nothing natural about it - that’s not how wood decomposes in Nature, and the mess of sticks has lots of air pockets in the soil where the roots would like to be. His experiment was only one year.
From further reading (I haven’t wanted to bother Dr. Johnson, and he can be very hard to get ahold of) it appears that the Johnson-Su compost process is long enough that the biomass passes through a full progression of seral stages, much like a forest ecosystem beginning with a clearcut does with plants - the first stage is “weeds” (hardy colonizers), followed by grasses and forbs, then sunlight loving fast growing trees, then shade tolerant species are able to sprout in the understory, eventually leading to dominance by “climax species” such as western red cedar (I first learned of this dynamic while participating in an old growth inventory study in North Idaho). The climax condition is quite stable and can persist for centuries.
In Johnson’s research he has tracked the species shift as the compost matures over a year’s time and found that in the beginning it is predominantly bacterial, shifting slowly to fungal (of the mycorrhizal variety) - as the detritus is broken down the saprophytic fungi diminish and the pathogens disappear in the beginning (consumed by bacteria) due to the lack of living plants. In the end there are thousands of species of viable microbes, however. Many are symbionts of specific plants, and it is the plants themselves that select them to partner with. Other fungi are generalists.
At about halfway through the seral progression the ratio of bacteria:fungi is approximately 50:50, which is what grasses and garden plants prefer. Mature forests like much higher fungal dominance. Weeds thrive at the other end of the seral spectrum. I think Dr. Elaine Ingham has a great table showing this.
If I was in your situation with limited options for processing the woody debris created by pruning trees and other vegetation management, I’d be inclined to chip it all and use it for mulch, or pile the chips to compost on their own. The material will follow essentially the same process as in the J-Su composter, it will just take longer and be less consistent. Top dressing with a thick wood chip mulch is good, too. The idea of that method depleting soil nitrogen is an old wive’s tale. If you can’t get it chipped, just pile it in smallish piles away from ladder fuels (in case a low burning wildfire comes through). Animals will use it for cover and it will eventually break down.
You never change things by fighting the existing reality. To change something, build a new model that makes the old model obsolete. R. Buckminster Fuller