I think there is a lot of conjecture going on.
It's free to get a quote from ETL or UL on what they would suggest you certify the device for. That can at least remove some of the research into what you need to do. I've had pleasant'ish dealings with ETL, and UL is... a nice logo to have.
If your power supply is class 2, note the stipulation, ie. low voltage protected from high voltage, and your power supply connects externally with a cord, you MAY not require further certification for your device.
It doesn't matter if the power supply is external or internal if the device has the equivalent of 15 watts or greater in the enclosure...
There will need to be flammability, impact, the "finger" opening, markings etc to be evaluated that will run tens of thousand dollars.
Might not be a completely correct statement. Low voltage devices don't have enclosure opening requirements as they are considered safe to be opened. Non class 2 powered devices do. There is a difference between an internally connected power supply and externally. There is also a difference if you product is certified for use with any class 2 power supply or is certified as a system. There are so many variables, without all the details of the project, it's all speculation as to what you can and can't do. I can say for certain, keeping the device low voltage will make the product easier to certify if that is needed and cheaper.
You can get a free quote (it's the only thing they will ever give you for free) and as them all the questions you want. Then you will know definitively.
Be very careful taking third party advice on regulations. They are country and sometimes even state specific. The consequences, in the way of liability, for not conforming can be scary.