Influence won't affect much of a battle. Only how many bodyguards your general has.
Otherwise, -1 influence isn't the end of the world. Retinue and other traits can help deal with that.
The only problem I can see is that your general will not get as many bodyguards as time goes on (the more influence, the more bodyguards that he generates overtime...if he loses some of that influence, if some of those bodyguards die they might not ever come back until he gains more influence to balance things out).
And then there's controlling settlements (influence helps public order), but that shouldn't matter that much - you shouldn't depend on influence to control public order anyway.
These sorts of trait depend on how you utilise your general in battle.
Hope I was of help.
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