I am making large batches of bread with 100% hydration poolish. In the current situation, using a large mixer and then having to clean the bowl after is not an option because of time. Mixing by hand does not effectively eliminate all lumps of flour. A dough whisk is not large enough for the task (think 10 pounds flour and 10 pounds water at a minimum). How do bakeries mix their pre-ferments when they can't use a machine?
Our local large bakery has mixers that have a tiltable bowl: mix, lift the bowl (with a kind of crane), dump the content out - wherever it's supposed to go. But we are talking about five to ten times the amount you asked about.
For 10kg in total, you don't need a mixer. Roll up your sleeves, wash your hand and give the poolish a good mix with your hands in a large tub. I have kneaded similar amounts of bread dough (60-70% hydration) by hand and know others who have done the same. Your 100% hydration poolish will be even easier to handle. Then scrape off what sticks to your hands and let rest as usual.
Note that a poolish doesn't need dilligent kneading, gluten development happens during the long fermenting phase, not during kneading. So if the poolish is somewhat "shaggy", it's ok.
I don't get your comment - Stephie describes doing it by hand.
At 100% hydration, a large, sturdy spoon should also work.
A large immersion blender might also do (no personal experience, I consider them more of a fad than a tool I need.)
Or if you have trouble dealing with effectively mixing 20 pounds of glop at once by hand or spoon, mix 5 4 pound batches and dump them into a bucket after they are mixed (or whatever size batch you find comfortable.) Or do those in quick succession in a small mixer and only clean it at the end.
On the further-out end of specialized tooling and timesaving that's not a mixer, something like a mortar box and hoe (the hoe has large holes in it), made of wood or stainless steel (wood seems more likely - I doubt you'd find one made for mortar that you'd want to use in food, so it would be a custom-job) - or a plastic tub and the "hoe." Of course that actually is a mixer, just one that's hand-powered.