No problem. I don't know what you know so I'll lay it all out.
First off, good move ditching the ACT. It's also known as Always Crash Technology. After you get the basics down it does more harm than good.
The 10x8 makes a HUGE difference. Even in stock form, with a little headwind and the 10x8 mine would almost hover. I personally only flew with the 10x8 because it performed so much better. You definitely need it with floats. I never had a problem running the bigger prop, only when I dunked the plane in the water and fried the electronics lol.
The old red version of the cub came with red stickers instead of blue and used older battery technology. The LP cub that you have comes with newer better LiPo battery technology hence the name LP Cub. If you bought the RTF Ready To Fly version that comes with the DX4 radio then you are using the latest version. It also means that the signal type the radio and receiver use is called DSMX. The old version was called DSM2 and the DSMX components are compatible with DSM2. More on that later.
The ESC Electronic Speed Control has a low voltage cutoff designed to do two things. First it protects the battery from being over discharged which damages LiPo batteries and prevents the charger from being able to recharge them. Second and more importantly, it shuts down the motor while there is still enough power to run the controls. This makes you land while you still have control of the plane. However, if you shut the throttle down and bring it up again it will give you a short burst of power and then cut off again, just like it's doing in your video. If you continue to run the motor past the cutoff point you run the risk of over discharging the battery. Once you notice a drop in power you are best off landing anyway but I find that I like to run to the cutoff and practice dead stick landings. The cub has such a nice glide slope that it really doesn't matter when you lose power as long as you are close enough to make a turn and put her down. I fly the first half of the battery all crazy and low and then the second half I bring it in closer and usually a few mistakes high and wait for the cutoff. Good practice for when my nitro engine cuts out lol.
Back to the topic....
The battery that comes with the plane is a 15C battery. That means that it can let out a max of 15 times the capacity of the battery, or 19.5 amps continuously. The nano tech can let out 32.5 amps continuous or 25C. The second number on the battery is a burst number, or short duration discharge rate.
What this means is that the batteries you have can produce roughly 19.5 and 32.5 amps without dropping voltage immediately. That's new, never used in ideal conditions.
The 1300 means that they can let out 1.3 amps total before being discharged. If you run full discharge on the 15C battery, you can 19.5 amps per hour with a capacity of 1.3 amps so your run time will be about 4 minutes. Since you don't use full power all the time and that motor shouldn't draw full capacity anyway, you can usually get about 10-15 minutes run time. This isn't all necessary to know, just FYI down the road.
Where this is important is when you draw more amperage than you should due a bad motor or esc. Then you exceed the amount of amps the battery can provide without dropping the voltage.
So, if your motor is shorted and now draws 40 amps (just tossing numbers out in the wind for example) then your batteries can't provide that power without the voltage dropping below the cutoff. The ESC will try to provide whatever the motor is asking for so it says give me everything and the battery can't do it so it shuts off. Without the voltage cutoff the motor would get hot, the ESC would be handing waaaay to much current and get hot causing the plastic to melt and it would eventually ruin your batteries. Even with the cutoff in place, the short burst of power will still cause a lot of heat in the esc and over time will melt the casing.
That's one scenario.
To check for this, you can install a watt meter inline and measure the amps coming out of the battery. That will tell you the demand. You can also take a volt meter and stick the leads in the back of the battery connector and run it to full throttle on the bench and see what the voltage is doing. If it's dropping significantly then you either have a bad battery or something else is wrong. If it does in on both batteries then you need to look at the motor and esc.
You also need to check all of your connections. I really should have mentioned this first. If you have a bunch of broken wires at your battery connection on the esc side, which is really easy to have since you have to stuff it into the battery box so tightly then even if everything else is working fine, you still won't get enough juice to the esc and it will see low voltage and cut off the motor. It would see it no matter what battery you use since it's on the esc side. The servos won't notice because they use such a small amount of power.
So...I'd check those things first, also look for any burnt or melted or unusually soft covering on all the wires in and out of the esc as well. All wires in fact. Those are signs of heat caused by too much amperage. Either high resistance (bad connection, broken wire) or a short (bad motor or esc, wires rubbed together). Let us know what you find or if you need more help in testing.
Should you consider upgrading the whole thing......
The DSMX/2 protocol your system uses is great. You'll get a lot of opinions on this but I use it on every single plane and heli I fly with no problems. Best of all you have half of what you need already with the DX4.
You would only need a new receiver, ESC and motor and with a little balsa or other scrap material you can have a mount, or they are available commercially. I have an article written on how I did my conversions, but that's for later if you decide that's what you want. It would be more expensive than replacing the motor, but if you have to replace the esc/rx (receiver) brick then it's a wash. You're better off upgrading if you have to replace the brick.
Hope that helps some. Let me know if you have any more questions and I'll do my best.