If you want to be really lazy about it, just get yourself a fat separator. Pour in the soup, the fat will rise to the top, and you can do what you want with it (i.e. dump it).
If you're reading this in an emergency, you can do this with just a strainer. You'll get better results if you chill the strainer before each skim, i.e. by rinsing it with very cold water. The fat will tend to congeal on the cold strainer the same way it congeals when it's actually chilled.
I've also heard that the fat will cling to certain leafy vegetables, like lettuce. If you have a lettuce head kicking around, try peeling off a leaf and dusting the top of the pan with it.
The purpose of using a roux, as opposed to just plain flour, is to improve the dispersal of starch molecules in the sauce. If you just toss a bunch of flour into a simmering sauce - or do the reverse, pour hot liquid onto dry flour - then you'll immediately start getting gross glutinous lumps and will find it nearly impossible to smooth them out.
Starch is not actually soluble in fat; instead the fat coats the starch molecules, making it difficult for them to clump together when incorporated into a sauce. Making a roux out of the flour (and making sure to give the sauce a really good whisking after incorporation) basically helps to ensure an even distribution of starch so that your sauce ends up smooth, not lumpy.
So to answer your question, the minimum amount of butter (or other fat) you need is however much it takes to fully incorporate the flour. If you've still got dry clumps, you haven't used enough fat. More fat is okay, but if you don't use enough then you'll suffer some of the same lumpiness as if you didn't use a roux at all.
I'm not sure if there's a magic number - it's going to depend on the specific flour you're using and the specific fat. Even if you narrow it down to, say, all-purpose flour and butter, it can depend on the brand of flour, the fat content of the butter, and whether or not the butter has been clarified (which, if you have the time and patience, it should be).
The 1:1 ratio (by volume or mass) gives you a pretty decent safety margin; you won't need more than that, but it's also not so much that it'll make your sauce greasy. But you really don't need to measure it out at all; I don't, I just add fat while stirring until the flour is fully incorporated. Sometimes I might be more generous with the butter simply for flavour purposes.
Bottom line, it's really pretty obvious when you haven't got enough fat. If it looks good, it is good, and after a few sessions without the measuring spoons, you should very quickly get a feel for how much fat you really need.
Best Answer
If you want to use the fat to keep the flavor, or because you are keeping a diet where fat is considered more beneficial than carbohydrates, there are several ways.
First, you can skim the fat, make a roux with the skimmed fat, then add it back.
Second, you can skim the fat, add a little bit of broth and an emulsifier, whisk until you have a nice thick emulsion, then add it back.
If you use a straciatella style thickening, the fat eyes will frequently disappear between the rags, especially if you can get them to break up very small.
But if you are asking because you want to save some work, I don't think there is a good way. Generally, if you use enough flour or other starches at the frying at the start of the stew (including tomato pastes and such), the fat will bind with the starch instead of floating. But it seems that your recipe is for a thin stew, and the fat continues to float there.