The OLED has the biggest, brightest screen. So I probably would get that. But there is no performance difference. So if the bigger, brighter screen doesn't matter, you can save money and get the regular Switch. There's a regular Mario Switch with a free game that's still available (maybe for a limited time).

https://www.target.com/p/nintendo-switch-mario-choose-one-bundle/-/A-88542077

Lots of games can be played 3 player on one Switch on the TV. There aren't many or maybe any 3 player games that are better when played on two Switches at the same time. You need the Switch and at least one more controller, but I would get two more. Some games can be played with one person just using the left or right joycon turned on it's side, but it's awkward and may be difficult for you or your daughter. Joycons with their small joysticks just may not be great for you or your daughter. (Maybe the third person can use the joycons...)

The pro controller is a great full size controller. The Power A nano is a smaller controller that may be a good size for your daughter. The Nano still has bigger joysticks than the joycons.

The Microsoft Adaptive controller does work with the Switch when docked, but it's expensive.

8bitdo has a couple of interesting controllers. The 8bitdo Lite SE is called an accessibility controller and puts all the buttons on the front and has separate buttons for the buttons that are normally in the joystick. (Switch joysticks have L3 and R3 where you press down on the joysticks for those buttons). So it may be good for you if the shoulder buttons are a problem.

https://www.amazon.com/Bluetooth-Gamepad-Raspberry-Mobility-Gaming-Console/dp/B0B4D3LC21

There is also the 8bitdo Lite that doesn't have joysticks. It has dpads that the Switch sees as the joysticks. If joysticks don't work at all for you, you might give it a try. But pushing directions on the dpad is always on or off. You don't have the option of pressing half-way like a joystick. So it will work better for some games than others. It also still has shoulder buttons. So it's not really sold as accessibility, but if joysticks are really bad for you, you might take a look.

https://www.amazon.com/8Bitdo-Bluetooth-Gamepad-Nintendo-Windows-Turquoise/dp/B07Y6N384Y

Just dance is one of the few games that can only use single joycons per dancer. But it can use phones as controller so you don't need a lot of controllers for Just dance.

A Switch Lite can work as a second device, but there are some real limitations to game sharing. You could use a cartridge on two Switches but not at the same time. You can use a digital game on two consoles at the same time with limitations:

https://www.reddit.com/r/NintendoSwitch/wiki/gamesharing

I feel like there are some down sides to the Lite for you. There are some joycon replacements that may be easier to use the regular Switch in handheld, but you can't replace the controllers on a Lite. The split pad pro joycon replacement are handheld mode only, but they have a full size joystick and may be easier to grip than joycons. There are other brands like binbok that also have bigger controllers that slide on the Switch like joycons but can work wirelessly too.

Any glass screen protector works great. Not really sure what to suggest for a case.