I know this is old post, but I have I think something valuable to add to this topic.
First of all, Julien, correcting background interferences is not Insane at all. If You have all large diffracting crystals (worse resolution) and measured substances contains many elements (30+) -- there is no way for some elements to have a interference-free space even for a single-background measurement (and situation is outstandingly worse for mentioned multi background method). Actually, this kind of interference correction (of background) at last is possible starting with Peaksight 6.5, and it works just remarkably (unfortunately it is not documented in release note - IMHO that is the most important improvement in Peaksight ever). After finding it out I started to use it fully on one of our machines (SXFiveFE), where we have all large diff crystals - that is like heaven-sent feature. Currently I have about 120 interference corrections for HFSE+REE+PbThU+MAJOR (about 43 elements). I was finding like 200 interferences initially, but it was creating interference correction loops and correctness of those in case of many overlapping loops is questionable. Thus some of mesurements had to move to "creative, self-correcting" mode. Everything is measured on alpha, no beta lines measured anywhere. Most of my measurements are with single sided single background as then there is less background interferences to correct (unless two bkgd can give more tricks for "Creative-self-correcting"). When using "exponential" with Peaksight there is a small catch: it will be linear if higher sin theta bkgd is more intense, so it is not solution to all problems as it can bring in some not-linear response in minerals with high variable composition.
Anyway, Peaksight was doing iterative interference corrections (of peak interference) I don't know for how long - stating "it's not doing that" is wrong. The only requirement stated for it to work in its documentation is to use Merlet's X-PHI. Talking about X-PHI, I find it very robust, in particularly that I see very good results with undervoltaged measurements (HREE+Hf,Ta,W lines on LLiF with 15keV beam), so that it does not require increase in voltage (classic approach, switching between 20 to 10kV), which preserves decent spatial resolution, and is in particular useful on FEG machine as it does not destabilize the cathode. I saw that NeXL has included X-PHI too. So there probably is a way to get that working in other sofware beside Peaksight.
As for negative interference (or bkgd interference) correction, the implemented way in Peaksight 6.5 works remarkably well. You can set background position even directly over other strong line of other element, even few elements, and that/those correction(s) will correct the intensity and measure it well. (I made few controlled tests to see that for myself). So background interference correction is not only NOT INSANE, but also a working reality.