Invasive tree roots- what are my options?

Hello! I'm not super new to gardening, but I am new at starting a garden from scratch... and I have a problem.
I caretake a property of substantial acreage, but only a small bit is usable space (we are on a very steep mountain, mostly brush). We've only lived here for a year, and so this spring decided to start designing our garden space. Yay!

The space we chose is the largest usable area on property, and is right behind our house. The sunlight is lovely back here. However, I think we were too excited to consider the terrible eucalyptus grove and Monterey Cypress roots nearby. Uhh... whoops. Classic case of thinking the problem won't exist if we don't think about it. Anyways, the trees do provide some shade for the garden, and assist in shading the house along with the further grove of eucalyptus trees. Our house has no cooling system.

When we created the garden space, we terraced the area with found wood, lined the bottom with a couple layers of cardboard, and backfilled with compost. The tree roots have already penetrated the beds and are taking all moisture out of the soil. Seriously, this last week my beds just won't stay moist. It looks like it's mostly the cypress roots, but on the beds farther away from the cypress it looks like it may be the eucalyptus?

Anyways, I would just like to hear some ideas for what you all would do, because I'm stumped!!! My thoughts:

My first thought was to cut down the cypress and the closer eucalyptus trees, but I do not own the property and want to consider all options before taking drastic measures. These trees would also be difficult to fall on this terrain. And this would also make my house warmer- I would plant trees and bushes to provide a bit more shade.

The cypress does provide some hill stabilization. Again, if I cut it down I would probably plant another tree in that area.

I need to cut a lot of the lower branches of the cypress anyways- would that actually slow the growth of tree roots? I honestly feel like Monterey cypress are so aggressive, it would just keep on going.

I've read a bit about folks digging trenches around their garden beds and putting in a metal roofing barrier.  like here Seems cool, but also a lot of work to dig in my clay soil, and not sure if it would keep those damn roots out!!!

What's up with hugelkulture options? Would that be a way to redo my beds, but keep the trees alive and the roots OUT? Hmm.

I really do not want to employ methods that require me to dig out the roots all the time. PITA. My partner and I started practicing no-dig in the garden we work at, and have seen a terrific improvement in the last couple years! Way less work for great results.

Hopefully I've explained my situation clearly, and here are some photos for reference. The best idea would have been to garden somewhere else instead of creating more problems to solve, but again- this is the only area I have to garden (The only other open/clear space on property is above the leach field...and closer to the owner's home). Even in this location we would have to terrace as we expand and minimize erosion.

I am desperate. My ears are open to any advice.

I'm pretty sure those big old trees own that space, and you're just passing through. What's the rule -- the roots extend half the height of the tree? Something like that?

You can dig down with a grub hoe and chop off the feeders, but if you're that close they will regenerate at light speed.

Realistically, I would be looking at raised beds with a hard barrier underneath, or container gardening. But that's okay -- don't be discouraged.

i guess this year, if you're already established, keep on mulching and consider watering. Otherwise, for the future I'm with Douglas, think about raised beds. You can make nice, no-dig hugelbeds!

Thanks for your replies! You are right. We have decided to transition to raised beds...
The upper eucalyptus will have to be cut down soon anyways, per owners request, but I think the cypress is staying.

So what would you recommend as a hard barrier underneath the raised beds? I've read about using a couple layers of landscape fabric, but I am worried about drainage, and many people say the roots will still find their way through after a year or two. It is going to be a bummer for the soil not to exchange nutrients and bugs with the natural soil beneath it.

I'm sure I'm missing a subtlety in your question but why can't you just build a hugel and start planting?  If you want a physical barrier, lay hardware cloth on the ground first. Sorry if I'm missing the point.

No problem, and thanks for asking. Hopefully this is clearer. Monterey cypress roots are extremely aggressive. I don't think hardware cloth or landscape fabric will keep the roots out for long, and while I love spending time in the garden- I would like to explore a set up where I spend that time enjoying it- not constantly fighting. I have been toying with the idea of a hugelkulture, but am still trying to understand the concept. I am not into the swales- we get way too much rain here in the winter and I'm on a hillside, so it would erode quickly.

So I could make a hugel as more of a mound on top of the soil, and could even make borders around it if I wanted to?...
Do you think the logs/branches underneath would prevent the monterey cypress roots from seeking out and stealing water from my plants? I would love to hear what you think about the hugelkulture method for this situation. These trees are serious.

I sat down and did some brainstorming today- I seem to be in between doing a method like that, or making raised beds slightly off the ground- then I don't need to dig out everything and lay barriers or dig trenches every year to cut roots or whatever. I am just trying to explore what would be best for my weird little space. It's not the most ideal spot, but if I can find a method that is a good compromise, I will be able to garden here for a few years. <3 The goal is to find a solution- that hopefully doesn't make more work or problems for myself in the future.