Interesting that the 130 grain was the weight of choice by Jack O'Conner and many others for so many years. The 130 grain in 6.5mm will have better ballistics.
No replacement for displacement. More volume in the case, more powder, more speed. At the top end, more doesn't get you much more velocity, just shorter barrel life.
From your description of the second round firing when you pull the slide back and let it go forward again. It sounds like the second round is not letting the gun go into battery on feeding, but the second impact from pulling back puts it into battery. That means you have not done the sizing and...
Velocity requires pressure from a given case. When you are getting 6.5 PRC/6.5-06 velocity from a case with 10 grains water smaller capacity, you are probably over 70K psi. Tread carefully.
The two have the same burn rate, but different densities. 7828 takes more volume for the same weight of powder. You could have trouble if you had a favorite max load of 7828SSC and tried to substitute 7828. The increased powder volume could cause excessive pressure.