I had ZERO hassles when I flew in for the last shoot from Seattle. I bought an SKB 2 gun case specifically because it's TSA approved, they asked for my key in Seattle and DFW, checked it out and relocked it. Total added time, maybe 5 minutes. I had a bunch of pellets in my other checked bag, along with my Chrony and even my belt knife, which happens to be a switch blade :-). Never got any questions on any of it. Pellets ARE NOT cartridges, and aren't treated as such. You could probably take them in carry on, but you can certainly put them in your gun case.
In the case of PCP's or CO2, no filled pressure vessels are allowed. Shooting my guns empty was the biggest hassle, especially since I'm pretty sure they never bothered to check, even though both rifles had gauges on them. BTW, rifles for that trip were my Cyclone and my S400. This trip, I'm probably taking 3, the Cyclone, my S200, and my Diana P5 Magnum. Those and my pellets will all go in the same case I used last time.
No trouble refilling your 410 from empty, Tommy, just be sure you cock it first. With no pressure in the tank, the hammer spring will hold the valve open unless it's cocked. Later.
Dave