Hola Kuwait340 -
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Sorry, I dont have anything specific about the CFM-56, and no experience in the A-320... I flew the DC8-70s with 4 CFM-56s long ago in the "last century", do not recall of a specific limit of tailwinds, but yes - I recall that we were supposed to be more careful starting with tailwinds than with the "regular" engines... There were no limitations in the AFM...
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I can however pass along some recommendations which I apply presently with the engines I operate, the JT9D-7Qs for the 747-200s. You can probably "translate" the recommendations and "convert" the info to some applicable numbers for your CFMs...
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Yes, in the classroom or the briefings, we insist on avoiding strong tailwinds for start, but in some cases, it is not avoidable -
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Try to "motor" the engine at the highest possible N2, but with more concern for the N1 you need to have - Normally, we need at least 20% N2 to move the fuel "ON", but if N2 is INOP, our N1 minimum is 4%... With a tailwind we insist on having N1 at a minimum of 5%, and I am more concerned of N1 minimum than N2... Keep a sharp eye on EGT rise (potential of hot starts) and monitor the N1 spooling-up to be normal (again, less concern with N2).
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Fact is - do not combine potential problems (APU with unsufficient airbleed start pressure with tailwinds) or ground airstart units which lack "vitamins" with a strong tailwind. Consider also amending which engine you start first, in case your airline SOP is to start the nş 2 engine first, with a tailwind coming from 4 or 5 o´clock relative to the airplane, I would start the engine closer to the fuselage, on the opposite side, possibly, the tailwind there will be reduced.
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This are the only recommendations I can give you -
Happy contrails