About 17 years ago I had a house built. The house was done but we could not move in because the septic was not done. Reputable septic installers were about 6 months out so I decided to do my own septic. Even though I had never even sat on a backhoe I bought a Case 580 CK backhoe.
I was required by the county to get the septic system engineered. So I did that. The system I had engineered was an infiltrator type system. This required, at a minimum, 4 feet of sandy loam for the drain field. For the 3 bedrooms and two bathrooms I needed to dig 4 ditches. Each ditch was two feet deep and fifty feet long. The ditches were 40 inches wide. The ditches needed to be flat and level within one inch over the 40 inch width and fifty foot length. I could not add material to make the ditches level and flat, all the earth needs to be undisturbed.
I had to make a blade that fit over the backhoe dipper that was flat. This blade fit over the teeth of the dipper. This made life much easier. I dug the ditches depth just slightly less than needed and used a shovel and rake to bring the ditches to final depth and level. I marked the handle of my rake with a Sharpie in 1/2 inch increments and used a laser that showed me the depth of the ditch so that I could use the rake to achieve the final depth and level requirements.
Though I had never even sat on a backhoe or done any kind of earthworks I was still able to install my own septic system. The county health inspector even brought other employees to see my system as it was installed. At first to show folks the wrong way and then, as I learned, what the right way should look like. When I was done the health department said my system installation was one of the best they had seen and employees were brought to my place to be shown what a proper installation should look like.
So, based on my experience, you should be able to install your own infiltrator type septic system.
Some things you should think about:
The drain field is not meant to continually absorb and drain away water. What it does is act like a sponge that collects water and then this water evaporates away. This is why, where I live, 4 feet of sandy loam is required. I don't know what is required in your area. Water flows into the ground in pulses, like from showers or from the washing machine. The water then evaporates away during periods between these pulses. Where I live, and I imagine this is how virtually all systems work, the water introduced into the septic system is meant to evaporate away. It is NOT supposed to continually soak into the ground because this would tend to contaminate ground water.
The septic tank is meant to hold solids. So the septic tank is always full. Mostly liquid. This liquid flows, in pulses, to the drain field. The solids are trapped in the tank. The pulses of liquid, water, flow into the infiltrator ditches, soak into the ground just beneath the ditches, then spread into the ground on either side of the ditches, and then finally evaporate.
Based on my experience you could indeed install your own septic system. Whether or not you get a permit is up to you. But you should understand how a septic system works and if you think you have the wherewithal to do the installation.
Eric