SLA batteries have never been good at providing continuous heavy current. A few seconds, like starting a car, OK. A few minutes, no.
As described, the problem sounds like the SLA not being able to maintain its voltage (OK, its a 12AH, but I didn't notice the voltage), either because it isn't fully charged or its not holding its charge or it just can't supply heavy current for more than a few seconds anyway. Or the load is more than expected.
Unless a lot of ballast is needed, NiMH packs of the same capacity are much lighter and far better behaved. In the old days, SLA were the only game in town, and were tolerated. Nowadays there are much better alternatives.
The ESCs have a built-in BEC to provide a regulated voltage to their own internal circuits and, via the red wire, to the radio and whatever is plugged into it. As has been said, it is bad practice to connect more than one supply together, so only one BEC in circuit at any one time. For testing by swapping ESCs about, it is better to use a separate battery for the radio and have both red wires disconnected.
Using a servo tester is also highly recommended. They don't just test servos. They can test anything that would look like a servo to the radio.