No, you can’t do that. As you noted, the Twin is a pair of DPDT toggles, but they are not wired independently, as I recall. The same pair of inputs feeds the center contacts of both toggles, so you can only connect one power supply as input.
If you are using a power pack and you use the reversing switch on the power pack to reverse trains on the rest of your layout, the same switch will also reverse the direction of both the turntable track and the turntable rotation motor.
I have one of these. I use a separate power supply and panel toggle for the turntable motor. I use DCC so the turntable track is wired to the track bus. As long as you’re careful about how you place the lead tracks with respect to the A and B polarity tracks from the turntable, that’s all yo need to do with DCC. For DC, you need to get your track power and run it through a DPDT, and then be conscious of polarity all the time.
OK. It appears that your turntable uses a split ring system to feed power to the turntable tracks. That is different to how the one I had eons ago was built, but that’s OK. Guys, correct me if I am wrong on any of what I am about to say as I’ve not messed with a split ring turntable in DC.
If the ONLY way you are feeding power to the tracks on the rest of your layout is through the outputs of the Atlas twin switch you are using for turntable track power, then you can leave the twin connected to the mainline and connect the turntable track power to the mainline. If you are feeding your layout through other feeders NOT done through turntable track outputs on the twin, then do not connect the twin to your mainline, but to the turntable track only. The key is using the center off position of the twin to kill power to the turntable tracks so you can use your power pack to rotate the turntable without the locomotive running off of it.
The current ones are split ring. All you have to do is jumper the mainline to the turntable track power as shown in the Atlas instructions. You cna either tap off the main, or just run a second pair of wires to the SAME side of the Twin that powers the mainline. This goes to the two terminals on either side of th emotr housing on the turntable - the turntable track power connections. The second switch on the Twin gets connected to the turntable motor wires on the side of the motor housing. That is all you should have to do. Put the mainline side of the Twin to the center off positon, and put the turntable side of the Twin either up or down and turn on the power. The turntable will turn but the train won’t move. If it is, you have one of these wires wrong. Do not overthink it, it really is as simple as it looks, there is nothing ‘hidden’ that only an electrical expert would be able to know.