From your post:
You do Taekwando, spend a lot of time training kicks/leg specific skills, and consequently have good sport specific endurance for those muscles.
Punches, which aren't a big part of the art, you don't do very often. Because of this, when you do do them, your arms get tired. Your solution, despite a very good template for success based on your experience building lower body endurance, is to do battling ropes, rather than finding something to punch. Mmm.
I don't quite understand where the bench press fits into all of this. Do you think if you added 50lbs and were suddenly, presumably, a 'good' bench presser, you would miraculously possess the shoulder/upperbody endurance you currently lack? Do all woes cease and virgins fall from the heavens at a BP of 300lbs - the sporting Nirvana?
If you want to get good upperbody endurance, specifically for punching you need to punch. Here's a program:
Equipment: heavybag & 16oz gloves
Program: Hit the bag - 3 mins on, one min off. Shoot for 150 punches a round. Build up to the point where you can do 12 rounds at this pace on any given day. Upper body endurance will no longer be your problem.
Note: This will not cure any technical deficiencies you may have, and will probably ingrain many that you do have. Ensure your technique is well drilled before you get too far along.