http://www.teatrskazka.com/Raznoe/Perec ... 35_03.html
This page lists the separate flame companies. Assuming that the "Backpack" ones are the ones to look at, I picked some of those I think the Red Army Handbook may be referring to, there are more up in the list being parts of "Fronts" and so on, but those do mostly Disband earlier, so I left them out (one of which has no Dates given at all).
Those are 20 in total of which 3 disband in early '43. There may be some missing due to me only looking at the "backpack" named ones only and leaving out the ones stated before. So my assumtion is that the numbers given in th Red Army Handbook are overall matching this page's list for those backpacked Flamethrowers, I can not reproduce it all to 100% tho, you got the sources I used if others see ways which make more sense, suggest them!
Handbook text and my Suggestion: I am not a scholar of Sovet Military, so if something is wrong/missing, just point it out/add it to this Thread.