Hi
EvilDragon's suggestion seems by far the easiest way.
Even though you mentioned Mint would be fine... In case you prefer more adventurous approaches, there seems to be some i915 hassle in booting live-usb-sticks, at least on Debian or Ubuntu. One of the following boot parameters might help (I'm not enough experienced to track it down, though): i915.modeset=0, i915.fastboot=1, video=efifb, video=intelfb, etc. It worked for me with i915.fastboot=1 on Debian KDE & LXQt but not on Xfce – don't take that for certain, it might well be coincidence. Once installed, none of those boot parameters are needed anymore. I (optionally?) added dmi_product_name=GPD-WINI55 to the cmdline in grub, however.
To get wifi working, I copied the content of [link to chromium googlesource brcmfmac4356-pcie.txt – I'm not allowed to post links] to a new file /lib/firmware/brcm/brcmfmac4356-pcie.txt. Not sure if that's still neccessary. As a reference: [link to archlinux wiki > GPD_Win > Built-in_Wi-Fi]
To rotate the display manager, I created a config file in /etc/X11/xorg.conf.d/ containing Section "Monitor", Identifier "DSI-1" and Option "Rotate" "right".
In Gnome, in case everything seems too big, probably gsettings set org.gnome.desktop.interface scaling-factor 1 helps.
For battery monitoring you need CONFIG_BATTERY_MAX17042 and CONFIG_INTEL_CHT_INT33FE, for charing CONFIG_CHARGER_BQ24190. In Debian kernel, they were added in 4.16. At some point (I think 4.20), the CONFIG_INTEL_CHT_INT33FE was lost. I'm currently on a custom compiled 5.3.9 kernel with Debian kernel config as a base and manually added I2C_DESIGNWARE_PLATFORM=y, CONFIG_INTEL_SOC_PMIC_CHTWC=y, CONFIG_PMIC_OPREGION=y, CONFIG_CHT_WC_PMIC_OPREGION=y, CONFIG_I2C_CHT_WC=m, CONFIG_EXTCON=y, CONFIG_EXTCON_INTEL_CHT_WC=m, CONFIG_TYPEC_MUX_PI3USB30532=M, USB_ROLES_INTEL_XHCI=m, CONFIG_INTEL_CHT_INT33FE=m, CONFIG_TYPEC_MUX_PI3USB30532=m, USB_ROLES_INTEL_XHCI=m. Not all of these are mandatory.
After charging or after booting from an USB stick, turning on my device sometimes results in nothing happening at all except the blue led glowing. For some reason, one of the very early bootloaders seems to get confused. Don't know if it's a a common issue or specific. In this case, Shift + Fn + F3 + Power and then waiting for some while helped. Although I'm not sure if that vulcan nerve pinch is rather esoteric – just powering and waiting long enough might work too.
Whatsoever, once you got it running on Debian, it's really neat!