I’ve toyed around with the idea of a Framework mainboard handheld for about a year, but on and off in design and planning it. I have a 3D printable design still a work in progress, but I’ve done tests with a few parts already.
What I have used is the 1135G7 mainboard, battery and a 7" 1200p screen from AliExpress. Use a HDMI expansion card to connect on one of the right side ports, with a slim FPC cable for HDMI connecting the display. This brings the total combined width to ~285mm. The biggest problem with the mainboard is its width- it is both too wide to have any room for a custom battery without making it ridiculously long, or thick (like a power wart sticking out from the bottom). I would like to keep it under 300mm long if possible, as close as possible to the size of the Steam Deck.
So, I just made it easier for myself and use the Framework laptop battery. It is very slim, and I have managed to flip it over and lie on top of the mainboard, for a profile under 15mm in thickness. Concerns about heat start to come up with the board and battery touching each other, especially since it covers the fan from one side. From my tests, the CPU and iGPU do pretty well and never went over 70C on loads. However, you do have to be mindful about not setting it flat against your table/desk because that will completely obscure the fan (it will throttle to 200Mhz when it overheats).
Additionally I have used a Teensy LC for a USB input device as the controller (haven’t added the thumbsticks yet). The Teensy and display are both powered using a second USB-C port.
Probably won’t go all in on trying the 3D printed design first, may just start a proof of concept with the parts sandwiched between two MDF panels.