The patient's weight doesn't affect the handling of the vehicle. Maybe if you were using the OLD ambulances from the 70s-80s that were modified station wagons and utility vans, but modern 'cube' ambulances are really trucks, and are absolute BEASTS. An extra 600 lbs on one side is NOT noticable at all. The main problems from transporting extremely fat patients aren't their weight, but their girth.
Inside an ambulance, you have a large shelving unit on the wall of the driver's side, which is at least 1 foot deep. It holds supplies, because often medics get slammed with back to back to back calls, and have time for little more than grabbing new linen and disinfecting surfaces between patients. They can go a full shift without running out of masks, bandages, etc., because of the storage shelves. The passenger side of the back has a large bench seat, often with cargo netting at both ends, and has 3 sets of seat belts. A medic can sit there while treating a patient, or the bench can be used to hold a medical transport team of a doctor and nurses (for example, during the transport of a neonatal patient in one of those rolling incubator things). The bench is additional storage for extrication and other rarely used rescue gear (most services have well-funded fire departments who will do that stuff, but the gear is there just in case). There is also the 'captain's chair'; a large, rear-facing armchair that's at the head of the patient, and is by the control panel for the oxygen and suction equipment. It also had a reading light for back in the day before computers when medics had to fill out physical paperwork. AND there is often another fold-down rear-facing seat beside the captain's chair, for additional patient/crew transport. If you were at a mass-casuality incident, and had to transport multiple patients in each Ambulance, if the medic stands while holding the cargo netting and grab bars, they could transport 6 ADULT PATIENTS (with minor injuries) in the back.
The fatal flaw of the system: the stretchers. They used to be mechanical, which meant that you had to completely lift them off the ground and take all the weight off of the undercarriage to pull up the wheels and get it in the truck. Usually, only one lifter is required, but if needed, there is space for more responders to assist. Patients who were small fats could be loaded by two medics. After getting the head of the stretcher and the two fixed wheels on the floor of the ambulance, the lifter grabs the bar at the end of the stretcher, and partially 'curls' the bar. Then they open their right hand and extend their fingertips to get the latch (it's like the second bar on the handles of self-propelled lawnmowers). They squeeze the two together while lifting and taking the weight off of the wheels. If they can't get it high enough due to being short, they can slightly go under it with their knees bent and use their quads in the lift, or if they only need an extra inch of height, they can "shrug", using their trap muscles to lift it. The other person lifts the undercarriage and gets the next two wheels rolling onto the ambulance floor, and then the stretcher is guided into the "bull horns" (part of the locking mechanism) as the final 2 wheels roll in. The stretchers themselves were like, 100lbs. Small fats are awkward, but even a ... oh, fit and strong 5'3" female (just as a random example) could lift a small fat (300ish lb) patient themselves. In therory, just having a second lifter can increase that patient size to 600 lbs. You don't WANT to have to keep doing that, as it's hell on the medic's body, but you CAN...
So here's where the problems arise. The stretchers had maximum capacities of like, 350lbs INCLUDING GEAR (defib, 02 bottles, etc). Deathfats more than max out the weight, which means that the undercarriage lock can give at any time, causing the entire stretcher to collapse. The only way to use them would be at their lowest setting, and pulling them around like a sled. So, no problem, you sled them around and get a 4-man lift to load the stretcher into the truck. Oh, their fat hangs over the narrow stretcher and you rip the FUCK outta their skin against those cabinets when you load them. Okay, no problem, just get a bariatric stretcher with hydraulic lift and more width. Well, damnit, now the stretcher won't engage with the locking bar and doesn't fit into the bull horns. Okay, no problem, we'll replace the locking mechanism with the beefy one for the larger stretcher. OH FOR FUCKSAKES, now the stretcher won't FIT inbetween the cabinets and bench seat. Okay, we'll have to MODIFY the seating and storage inside the vehicle to accomodate this new stretcher.
And voila, you now have a bariatric ambulance.
The actual "lifting" of the patients is as much of a shitshow as you are imagining. There is no professional, smooth-looking process, as there are an infinite amount of ways to get 'stuck' inside of a buiding. The basic rules are keeping the load close to your body, several small lifts instead of one long one, no twisty-motions, and don't injure anyone in the process. Often the patient gets stuck in a tub (where they are wet and slippery), wedged between the cabinet and toilet, at the bottom of a steep flight of rickety stairs, in an apartment building with no working elevator, etc. It's just lots of people, and improvised tools like lifting straps, repurposed seat belts, towels, sheets, scoop stretchers, 'stair chairs', and anything else you can imagine to get the person out with at least some sort of dignity. And it's physically painful to patient and responders alike.
Well, unless they're dead. Then it only hurts the first responders. Still, dignity is important - which causes more pain and risk of injury to responders. Plus, depending on how long they've been dead and how 'fresh' they are, you're either dealing with rigor which makes them hard to 'fit' anywhere while you're moving them, or they are very.... um... "soft".... and you worry about goo and rupturing. Usually by that point, it becomes the coroner's problem, and I don't know how they do it - though I'm sure it involves plastic sheets, nylon tarps, and lots of manpower.