How do you get the rancid smell of elderly diarrhea out of clothes?

Bonfire in the backyard.

Buy lots of pants and throw out all feces stained ones. It's going to be expensive but cheaper than replacing your washing machine.

If this is going to be a regular occurrence dad will need to be less picky about his clothes.

The first product you need to buy is OxyClean washing machine cleaner. Washing machines have to be regularly cleaned on a scheduled basis or clothes will stink that are washed in them.

I used to use a detergent called '20 Mule Team' borax soap powder with some Tide powder mixed in then do a double rinse.

If a wash is extra nasty, fill the machine with the hottest water it can. Add the 20 Mule Team and the Tide. Let that dissolve. Soak the load for several hours or overnight. Then drain the water, add the soaps again and put the wash cycle on - extra hot. I find that nothing worked as well as this.
I was an in-home caregiver for 25 years, so I've tried every combination of every kind of detergent.

If you have a newer (he) washing machine with no agitator, you can't use the powdered detergents directly. Dissolve them in a half-gallon of hot water first and you can.

For things like towels, sheets, and chux pads I always went Tide powder and old-school bleach in hot water.

I feel after a couple of washes if the odor is still there the clothes need to be thrown away. Do not put in dryer until the clothes are completely clean. Feces contaminated clothes should be washed right away and the washer sanitized with vinegar & baking soda.

Is Dad wearing disposable underwear?
Buy Dad replacement clothes from the thrift store. It will be much more affordable.

I had no luck with vinegar....I AM having some luck with cutting a lemon in half, and throwing a full box of baking soda into the load WITH the clothes first. I squeeze the lemon, and then place the two halves into the mesh bag and close it - and throw it in the washer along with emptying a full box of baking soda...That's the first cycle.
Second cycle I rewash with regular laundry pods, (leaving the lemons in the washer/mesh bag) and run it through again.
Been doing that for the last couple of years and that seems to do the trick.
Hope this helps.

How about using Pooph spray?

I agree with " toss the pants". Perhaps replace with less expensive sweats , something that can also be tossed out if needed w/ o forfeiting too much investment.

Speak with your father and his care team at the facility to be sure that he is being provided and, is complying with use of appropriate incontinent supplies.

Spare yourself and your washing machine. Toss them and buy new ones in bulk. By the time you buy products to kill the smell and stains, and run a wash cycle so many times, you’ve spent as much time, money, and effort to negate the cost of new ones. Amazon or WalMart sells in bulk.

There’s situations to pinch pennies; this is not one of them.

It took my husband a full day to take our washing machine out to the garage, take it apart to clean it properly, reassemble it, and bring it back into the laundry room. I can't tell you the sludge we found in there, it was mind bogglingly disgusting. And we do not wash poopy clothing. Only heavily soiled work clothes covered in dried mud mostly scraped off, which is why I have a heavy duty washer. Our clothes were coming out of the washer smelling like old sponges. "Washing machine cleaner" products DO NOT remove sludge that is trapped between the washer barrel with the holes and the unit that HOLDS the barrel.

Throw the clothes out and buy new ones, it'll save you a giant, nasty headache in the long run.

I find that doing a pre hand washing and soaking has helped. I never put in washer soiled. I spray with oxi clean and soak. Then I add soap give a quick hand wash and then wash out in washing machine.
Sounds like you now need to get a cleaner for your machine