I agree with what Prof. Manary has written. It seems that there are three important things to focus on.
The first is infection -- if the pneumonia is not improving with typical antibacterial coverage (ceftriaxone +/- cloxacillin might be a reasonable choice to start with), then you must think strongly about TB. Make sure to check HIV status also. Remember that GeneXpert has a relatively low sensitivity in young malnourished children so you should have a low threshold for treatment even if the test is negative.
In terms of dehydration, if there is a concern for diarrhea losses, then be sure to continue to provide ReSoMal (or ORS if you do not have it available) to account for stool losses -- typically 50-100 mL per loose stool is needed to help compensate for fluid losses. Use an NG tube if you need to.
In terms of nutrition, it does seem like F75 will still be necessary. You can use smaller and more frequent dosing if needed due to vomiting and diarrhea -- for example, take the total daily dose and divide by 12 and provide this amount every 2 hours. Or even divide by 24 and provide the amount every 1 hour.
If the child is drinking by mouth, one trick I have used is to mix the F75 with ReSoMal to dilute it even further and make it "thinner" for the child to drink with less difficulty. (Of course be sure the child still gets enough F75 over the whole day.)
If you are in a place with access to an infusion pump, you could in an extreme case have the F75 run as a continuous very slow NG infusion over 24 hours.
On the whole, as the gut mucosa recovers and the brush border enzymes (disaccharidases, etc.) recover, the diarrhea will start to slow down. If the diarrhea has been persistent for some time, consider other GI infections -- protozoa or opportunistics due to HIV/malnutrition.