Hi Piotr Cerba,
Because kl43 have the ROM bootloader, you don't need to install the bootloader application at first.
Please refer to the doc:Getting Started with the Kinetis ROM Bootloader.pdf, this document is for your KL43.
1. check your COM in your computer management:
This picture is from my side, my com is COM33, this step is very important!!!

2. Open a command prompt in the directory containing blhost. For Windows OS, it is <install_dir>/bin/win.

3. Ping the COM port in blhost

If you get the current version, you will pin the COM successfully.
4 Erase all flash, and write the KL43 application bin file

All these steps you can find it from Getting Started with the Kinetis ROM Bootloader.pdf, please try it again, about the FRDM_KL43Z4.bin, you can find it from :FSL_Kinetis_Bootloader_1_2_0\FSL_Kinetis_Bootloader_1_2_0\apps\binaries
You need to copy FRDM_KL43Z4.bin to path:FSL_Kinetis_Bootloader_1_2_0\FSL_Kinetis_Bootloader_1_2_0\bin\win before you write it to your chip with UART ROM bootloader.
It works on my FRDM-KL43
Wish it helps you!
If you still have question, please contact with me!
Best Regards,
Jingjing
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