I’m adding a second answer to address frame, suspension and drivetrain. I originally took your “underbody” to mean just sheet metal body parts and addressed only that in my other answer, but you’ve clarified now and you mean everything under the vehicle.
This video shows a professional grade needle scaler in use with an adequate air supply, and it gives you a good idea of its capabilities. Cheap-o needle scalers or those powered by a portable compressor are too slow and wimpy for a large job such as yours.
Once the loose rust is removed, a rust converter or encapsulator followed by rust preventive paint gives good results, even if the finished result is not perfectly smooth.
Here are before and after photos showing some of the frame and suspension.
The bright white stuff on the fasteners and adjusters is so-called “liquid electric tape” that I brushed on to prevent further rust. I didn’t want to leave threaded parts as bare metal and have the threads rust away, but coating them with the black polyurethane paint was not an option because of how tough the paint is. The white stuff comes off quickly and cleanly with a wire brush yet still provides adequate rust protection.
I went through this process on the entire underside, including the frame, suspension, steering and brake parts, differentials and axle tubes, drive shafts and the cast iron parts of the exhaust system. The engine itself didn’t need remediation except for the oil pan, which had sheets of rust coming off.
This was a 2009 Dodge Ram 2500 diesel 4x4. I had been warned in late 2017 at my annual vehicle safety inspection that my truck would be failed for rust within the next couple of years if nothing was done, so I undertook to fix it in July/August of 2018, summer here.
I need a heavy duty truck with a snowplow because I live in a heavy snow area at the end of a long driveway. Commercial snow removal service for my property would be ridiculously expensive. Before I decided to tackle the rust myself, I priced out the cost of a replacement truck by going to a local Dodge dealer and telling them I don’t need a truck anymore and I’m not buying anything here, so make me an offer on my truck, I’m selling it, not trading it in. It was clean, had low mileage, everything worked, and even with all the rust underneath, after a mechanic looked at it they offered me US$28K. (US$1.00 = NZ$1.60 today.)
A new truck similarly equipped was in the range of US$60K, so a replacement would be on the order of US$32K. I prefer to buy new and keep forever. So I had a very strong financial incentive to fix the rust and keep this truck on the road for the full 20 years (minimum!) that I like to keep a vehicle.
Unlike you, I was already set up with everything I needed to do the job, most importantly an extra tall vehicle lift in a garage with a 15 foot ceiling so that I could stand fully upright while I worked and adjust the vehicle height at will for comfort whether working on the bottom of a rear leaf spring or the top of a wheel well. And I already had a powerful stationary air compressor.
But the project took two months! I worked at least 4 hours per day for more than 60 days in a Tyvek bunny suit with eye, hearing and respiratory protection. It was brutal! I swept up and saved all the rust flakes, chunks and dust every day and in the end it came to 33 pounds (15 Kg).
Although the rust converter and paint system worked well, I have not named it and I won’t recommend it because other products have worked as well for me on other vehicles without the myriad problems that I experienced. I’ll explain. This polyurethane coating requires two coats, and timing is critical. The second coat must be applied after the first coat has dried to the touch but while it’s still tacky. If you let it dry fully to a hard surface, the second coat just beads up like water on wax.
Complicating the timing is the fact that this paint cures by reacting with moisture in the air and the curing speed depends on humidity. At very low humidity, you get a couple of hours of till the tacky stage, but in the jungle-like summer humidity where I live, it’s tacky in under 30 minutes and fully cured and no longer amenable to a second coat in under an hour. That also means that your brush gets glommed up with cured paint in an hour or two when it’s humid.
I overcame that by running both heat and air conditioning at the same time, which dropped the humidity down into the mid-20% range and gave me a couple of hours of working time. Each painting day I would start a new section, work two hours on it, then go back to where I started for the second coat and cover everything in the same order for the next two hours. If I failed to finish second-coating anything, I had to scuff-sand it the next day to keep the second coat from beading up.
A new brush was needed every day. Ordinary paint thinner doesn’t work on this paint so you have to buy the manufacturer’s own special thinner, which is very expensive. It’s cheaper to just throw away your brushes and start new every day.
And speaking of thinner and solvents, this stuff is hazardous to one’s health. The solvent includes both aliphatic and aromatic naptha and other nasty ingredients. It’s sold to the general public, but it’s labeled for professional use only. It requires serious respiratory protection to prevent brain and liver damage, so I used carbon VOC filters and a tight fitting silicone rubber respirator mask.
But I must say, it’s been nearly five years and I still have no rust to speak of so I accomplished my goal of extending the life of this vehicle. At the cost of spending a significant fraction of my remaining life on the project! (I’m only a little younger than you.)
Instead of this polyurethane, I recommend Corroseal as a rust converter followed by an oil-based rust preventive paint of your choice such as Rust-Oleum. (I have no connection with these brands, but I have used them.) Corroseal is a water based translucent emulsion that turns rust black and prevents further decay. It dries transparent and acts like a latex primer for the top coat of oil based paint. It’s benign stuff. There are no timing issues on the finish coat. You can apply the Corroseal everywhere at once, then follow up with the oil based paint at any time. Although this system is not glass-hard and may not be as durable as the polyurethane, it has given me good service on another vehicle on the same salty roads.
Now your project: You don’t have a lift or a pit. You could make a safe ramp and use a mechanic’s creeper. Not sure if you have the monster compressor you need for either sandblasting or needle scaling or if you’ll use a smaller portable compressor and pause every time the pressure drops below a usable level; if so, you’ll spend most of your time waiting. The tilting car scenario? You have my best wishes.
Whether sandblasting or needle scaling is your method of choice, you’ll still spend over 200 hours on your back under the van if my experience is any indication. Will the inspectors pass the vehicle after all that or will they find another excuse to fail it?