I finally received my CC debugger (a cheap knockoff from AliExpress) and immediately set to work flashing my CC2530 module: success!

For those interested in the process, I followed this documentation on using the TI SmartRFR software to flash the module. The firmware I used was based on Z-Stack 3.0.2, built following this tutorial I mentioned before (you can download the hex file from the Files section: CC2530ZNP-with-SBL (Z-Stack 3.0.2).hex).

To check that the flash was successful, I used the TI Z-Tool software (don't forget to check "Scan all Enabled com ports (startup and manual scan)" in Tools > Settings > Serial Devices > Auto Scan). 

While waiting for my CC Debugger to arrive, I looked around on the Internet for possible reasons why my previous attempts might have failed. Although I didn't find anything directly related, I decided to take a closer look at my components and found out the solder joint of D5 on my Wemos D1 mini was cracked... A quick rework and I successfully flashed the firmware using CC loader library (for those interested, I also uploaded the bin file in the Files section).