I think you've got all the numbers you need, except for the pin weight.
You know you can't go over 17k on the trailer, or 23k combined. Can't be over 6,100 on rear axle, etc. In your case, GCVWR will be around 15,700, because 23,000 - 7,260 = 15,740. Seems your trailer can handle that okay. But your truck cannot.
You've got 2,340 of remaining GVWR capacity. You've got only 1,780 of remaining rear axle weight capacity. Because 15% of 15,740 is 2,360. Too much for your rear axle if you must follow the ratings.
Since it is a flatbed, your pin weight can vary widely based upon how you load the trailer. It could even be zero, or a negative number on the pin if you loaded badly behind the axles. Don't do that, it won't tow well. Shoot for 15% of total load on the pin as a minimum for safe handling. Plan to cross a scale before you get out of town, and be prepared to shift the load if the numbers don't work.
I think you're looking at a max trailer weight of about 12k if you can load so that 15% is on the pin (1,800). 12k minus your trailer tare weight of 5,600 is a net cargo weight of 6,400. Might be asking a lot to load that perfectly, so reality might be a bit lower than that.