1. EC 261/2004 compensation is payable by the operating carrier and the operating carrier only. It does not matter which carrier ticketed the passenger. Thus, on this KL-operated flight, any claim is properly made to KL.
2. According to OP, the flight arrived 3:55 late. For a Type 3 flight, a delay of that length may entitle OP to compensation of EUR 300. (EUR 600 reduced by 50%). One is always free to accept alternative compensation.

Claims agencies do not accept matters they do not believe that they can win because they do not get paid if they lose. Thus, they have an incentive to get it right. Here, it appears that they did. The delay of roughly 2 hours was not extraordinary but also did not result in an arrival delay of greater than 3 hours. The de-icing delay is certainly a weather-related and extraordinary matter. The fact that it would not have occurred but for the earlier delay may be true, but does nothing for a claim under the Regulation. Prevailing over a KL rejection will mean fighting a long battle (which KL may well want to pursue because it does not want to give up the precedent of such delays).

I suspect that someone at KL hit the "send" key too fast. While it appears unfair to OP, the fact is that the payment to the friend was likely an error with which KL is stuck.

OP may certainly try other claims agencies or could, of course, pursue his own claim without an agency, but the underlying factual problem here is likely to mean that this all fails sooner or later.