Lacto like warm temps ~48c, you will certainly want to hold and maintain that until you reach your maximum acid level which will be 1.8-2.2%. Holding that temp will also kill any other yeasts that you don't want.
Yeast survive at that temperature. They don't reproduce well above 100-105F but cells do not die. Once the wort cools in the fridge the yeast will resume reproducing and fermentation.
Its as simple as making some starter wort, and throwing a handful of uncrushed grain on top. Hold at 48c for 3-5 days, the grain will sink and you can just pour the gut off the grains. You could easily make 2l use 1l and put the remaining 1l in a sealed mason jar in the fridge for later. Your acid % with this method will be about 1.25%, but its easy enough to titrate and figure it out for sure.
This is a fine process for what you are doing because you use such a small amount of soured wort in a batch; however, it is a bad beer waiting to happen at a larger scale. Using luck of the draw to culture off grain increases the probability of off flavors in the beer because the volume of off flavor compounds is far greater in a full batch soured in this method over five percent or less that you use. That's not to say one cannot ever get a good sour beer out of this process but the probability is not great. That's why most brewers kettle souring use selected lactobacillus strains.