There are probably hundreds of exceptions to my thinking though Every situation is different. But generally, my thinking starts here:

If you are a rank and file employee and you don't have big aspirations for climbing the corporate ladder. Then go for it. It's just a game of both sides taking what they can from the other. You certainly don't owe them anything and they owe you nothing (besides what's in the company manual). If they allow a man to take 6 weeks off when his wife/significant other gives birth, then take it. I still think its wrong for the OP to double dip, but we are now just talking about paternity leave generally.

On the other hand, I don't know any executive/management types (or "on track" employees) that have taken, or would take, 6 weeks for paternity. Ignoring what "rights" they have under the general company policy, it's just not needed. There is no real justification for it. If a guy gets sick, needs surgery, has a family emergency, etc., then sure. Take the time you need. There are things way more important than work. But watching your wife have a baby is not one of them that should translate into 6 weeks of uninterrupted, no responsibility, time-off spent at home. A few days, a week . . 2 weeks? Yeah, sure. 6 weeks? No way.

If you are good enough at your job and your position is important enough at the company, then taking 6 weeks off just doesn't work (or else places an enormous unearned burden on your peers at the company). If it did work, then the company probably doesn't need you anyway. Your peers should have your back for real emergencies and issues. I don't think you should make them cover you for 6 weeks for paternity.