An update on what I have tried:

#1 converting gcd file (nuvi16xxEU_regionfileonly__300.gcd) to rgn file with RGN_tool
rename the file 113201000.390.rgn
drag it over the Webupdater.exe
It ran, but the device could not been updated.

#2 Rename nuvi16xxEU_regionfileonly__300.gcd to gupdate.gcd, place in a root folder named Garmin on a microSD
Could not boot, freezes at Garmin logo.
drag the 113201000390.rgn (I converted myself) over the Webupdater.exe
It ran, but the device could not been updated.

drag the 113201000390.rgn (downloaded from Neil) over the Webupdater.exe
Loader loading appeared on touchscreen
but later Software missing appeared later.
Message from Webupdater:
Your Garmin device has been updated

There may be additional updates available for your device, such as voice updates.

Click Next to check for more updates.
Click Cancel to quit.

After clicking next:
There was an error communicating with your Garmin device. You should try again after unplugging and plugging back in the device and ensuring that your Garmin device is on.

If the problem persists, contact Garmin Technical Support.

#3 In case the Webupdater.exe is the wrong tool I used, I try the same procedure with updater.exe downloaded from Neil

drag the 113201000390.rgn (I converted myself) over the updater.exe
It ran, however later I got a pop up box saying:

The Garmin device is unable to program the requested memory region.
This could be due to an incompability between this application and the Garmin device.

If this error continues to occur, please contact Garmin Technical Support.

And the text on the touchscreen:

Loader Loading

After that Software loading.

Now, I tried with the 113201000390.rgn provided by Neil,
It ran and I got the pop-up window saying The update was successful

The touchscreen was showing loader loading, but later software missing.

The software missing stays on the touchscreen after unplugged from the pc.