Thank for the datapoint - Earlier versions of the ExtIO_LimeSDR.DLL didn’t support the Mini because even I tried to see if it would function and it didn’t. Good to know that this 'Tv2 version is supported on both flavors of Lime now.
73 de Marty, KN0CK
Yes! I found how to solve the problem!
Some tricks with Windows registry, and it works 
I don’t know will it work in other versions of Windows, but on W7x64 it works fine!
So, steps to do:
Open Registry editor and make search of string “HDSDR”. You will get some results, but you need something like
[HKEY_USERS\S-1-5-21-594861094-2120942956-3325901376-1001\Software\HDSDR]
This long number between HKEY_USERS\ and \Software should differ from mine. You have to copy-paste this YOUR numbers to notepad.
Now create empty text file, and copy this text into it:
Windows Registry Editor Version 5.00
[HKEY_USERS\XXX\Software\HDSDR\ExtIO_LimeSDR_1.05.dll]
“000_Description”=“HW Name”
“000_Value”=“LimeSDR Mini”
“001_Description”=“Channel”
“001_Value”=“0”
“002_Description”=“Antenna”
“002_Value”=“1”
“003_Description”=“LPF bandwidth”
“003_Value”=“20000000.000000”
“004_Description”=“LPF enable”
“004_Value”=“1”
“005_Description”=“Srate index”
“005_Value”=“5”
“006_Description”=“LNA gain”
“006_Value”=“13”
“007_Description”=“TIA gain”
“007_Value”=“2”
“008_Description”=“PGA gain”
“008_Value”=“19”
“009_Description”=“Current Device index”
“009_Value”=“0”
“010_Description”=“Calibration bandwidth”
“010_Value”=“20000000.000000”
“011_Description”=“Current LO frequency”
“011_Value”=“935000000”
Now replace XXX in [HKEY_USERS\XXX\Software\HDSDR\ExtIO_LimeSDR_1.05.dll] by your string you copy-pasted in step 1.
Save your file as ExtIO_LimeSDR_1.05.dll.reg somewhere in your HDSDR folder.
Double click on saved file, and them ok. (Usual procedure to add data to registry).
Start HDSDR with Lime EXTIO (don’t forget to connect your Lime to USB).
Take beer and enjoy listening to radiowaves.
@Armin
While this is nice workaround it doesn’t solve the problem that it doesn’t work below 30MHz.
Can you set SXR to 30MHz in LimeSuite and tell me what happens? It’s near Calculate button at SXR tab.
And if possible do it again after HDSDR fails to set it below 30MHz, connect the Mini board to LimeSuite and don’t press Default button.