The fluid you need is JWS3309, usually cheapest at Toyota dealers.

Check the fluid color and level using the yellow dipstick. It is under the air intake tube just behind the radiator. Be careful, the plastic end can break off if you bend it.

I never knew that it didn't have a drain plug. Have a look around underneath, it is the same size as the oil drain plug.

If there is a drain plug, replace with a new crush washer the same size as for the motor oil - aluminum. Drain out the 3+ quarts of cold ATF (do not run the engine first as you want the sludge to be at the bottom of the sump). Then, using a long neck funnel, add the same amount of 3309 through the dipstick tube. Warning, it is a slow process and cannot be made to go faster.

You are correct, using a machine to flush the ATF is a no-no. If you wish to flush after dumping the sump and refilling it, the top cooler hose is removed and inserted into a clear 1/2 or 5/8 line and directed to an empty gallon jug, marked at 2qt levels (one mark). When the lines are hooked up, start the engine and let it idle as the fluid is pumped out - a fast process, careful. When 2 quarts has been pumped, turn the engine off and add 2 quarts of 3309 to the tranny. Repeat until you get the desired color.

Some folks just drain the sump 3 or 4 times within a week to get the same result.

It might be helpful to remove the fresh air intake hose to the air filter, lots more space to work in.