First posting for this new guy, so here goes.
I have an MTH/Rail King Southern Pacific 4-8-4 (X4449) loco and tender with Pro Sound.Exact model number from the box end flap helps (30-XXXX-1 typically). They made tons of this model but different variations so it does matter in technical discussions.
Can this train be powered by a Lionel transformer and if so which ones ? I currently have a Lionel CW 180 (with power brick) I think you mean either the GW180 (given you said external 180Watt brick) or possibly it could be a CW80 and lionel did make a similar one with an external brick. Again, being exact on model number can be important in a technical discussion because Lionel made a lot of different products and there are technical details that matter when asking for help.
Seems I read someplace that the Lionel train transformers don't play well with MTH trains, but Lionel trains will run fine using MTH transformers. I'm one of the people who often say that from experience. There are 2 main reasons or facts that come into play. The typical Lionel GW and CW series of electronically controlled transformers use a microprocessor chip between your button inputs (bell, whistle, and direction, as well as the throttle pot handle) resulting in a slight delay of the exact moment when you press a button and the resulting AC waveform signal is modified carrying that function. Same delay happens when you release the button- the signal continues to be output for just a split second. MTH PS1, PS2, and PS3 engine will operate under conventional control (varying track voltage and bell and whistle/horn DC offsets in the AC waveform) however, the advanced commands that require precisely timed short sequences of bell and whistle/horn can be trickier to get the correctly timed pulses to be read as a "function" rather than just random blowing the horn/whistle or activating the bell. That's again a very specific function, so sure, you can "run" the train, but do you get all the features and functions- especially something that sometimes needs done- the reset command. Further, since these transformers are modifying a fixed 18V output to vary the track the voltage, they are not producing a pure sine wave power to the track. This especially matters in MTH PS1 engines- because they are conventional only with advanced control features, so they must be able to differentiate signals from smooth AC power and thus do not like and may not run on chopped or distorted power from such electronic controlled transformers.
And you might ask yourself- OK, so MTH transformers can run either MTH or Lionel. Yes, true, because MTH implemented the control using more analog components in the case of the modern Zcontroller, not a microcontroller. They also designed the circuit to attempt to smooth the AC power such to be more friendly and more sine wave like. So specifically a Zcontroller tends to have near instant response for bell, whistle and direction, and more friendly smoother AC power. I'm not saying it's pure sine wave and some earlier MTH PS1 still may not be happy as they improved the MTH PS1 firmware over time to be less picky and sensitive compared to the earlier versions. Again, the bottom line. MTH PS1 is a special case and depending on the engine and when it was produced it can be very picky about what transformer even works.
And finally, does the battery in the tender need to be there to run the train ? Yes, because MTH PS1 uses microprocessor for both control and sounds, and the battery keeps it running when you drop power for direction change. Also MTH engines start up in neutral and must be powered for at LEAST 20-30 seconds because there is a firmware wait or pause during the startup before you can switch to forward. This also charges either the battery or capacitors during this extended startup time. Again MTH must have a working battery or capacitor backup system working, and functional providing power to the microcontroller or else bad things can happen and ultimately you will not be able to run the engine.
So with that- MTH engine new to you, it's generally a good idea to open the shell, and identify the battery and if it is an original white one- immediately change it.
Again, if you see this white battery- no matter how good you think it looks, how brand new in the box it is- stop and replace it.
Any help on this would be appreciated.