The sentence as it stands is too wordy, and lacks punctuation. A covering letter needs to be crisp and to the point. Besides what do you mean by 'good customer-orientation'? Never use words simply because they sound sophisticated. Experienced business people and senior academics can see right through that habit, and it will score negatively. 'Orientation' does not seem to me to be the best word you could use. Do you mean your 'strong customer skills'? I sense that you do.
Personally I find it wearisome, if, by the time I get to the end of a sentence I have forgotten what it said at the beginning, and have to go back and re-read it. This is what happened to me here. It would have helped if you had applied some punctuation. I would recommend breaking the sentence into two as follows.
'Our success in the consulting project has demonstrated my strong customer skills. Had it not been for my tactful and proactive approach, the client would not easily have accepted our cost-saving recommendations.'
If 'strong customer skills' does not seem right to you, how about 'sensitivity to the customer's needs'?
I have changed 'pleasantly' for easily, since business people are never terribly concerned about how pleasant people are, as long as they agree 'easily' with suggestions. Note that I have also corrected the tense of 'accept' to the perfect - 'have accepted'.