You can use a second Pico as your debugger.
https://therealprof.github.io/blog/usb-c-pill-part1/
https://github.com/probe-rs/hs-probe
It's getting harder to source these with stm32f103's as time goes on, so you might be better off choosing a stm32f103 dev board
Firmware: https://github.com/devanlai/dap42
https://www.nxp.com/design/microcontrollers-developer-resources/lpc-link2:OM13054
https://www.nxp.com/part/MCU-LINK#/
You can use DAPLink firmware with any of it's supported chips (LPC4322, LPC11U35, K20, K22, KL26). You'll need to use the 'develop' branch to use GCC to build it. You'll need to find a chip with the correct
Firmware source: https://github.com/ARMmbed/DAPLink/tree/develop