Reduce minimum operating voltage of an H-bridge
This solution is 'highly likely' to work.

Summary: 


Provide driver IC with an external 12V supply - see data sheet.
Possible bonus: The existing maximum 40V Vhv limitation is probably due to either the maximum allowed voltage of an onboard regulator IC or Vdsmax of the MOSFETs used - or both.
When Vdd is supplied from an xternal 12V supply the regulator can be removed and the board will then be able to be operated over the range fron 2V (probably) to whatever the MOSFETs allow.


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The webpage that you cite states "3. Working voltage 10V-40V".
Obviously, your 2-40V violates that requirement.  

The HIP4081A data sheet specifies a Vdd of 12V typical and 9.5V to 15V operating. The supplied board needs to get this voltage from somewhere. 

The board contains a horizontal TO220 package device. It is likely that this is a regulator which derives Vdd from the high voltage H-Bridge input.
The 10V Vhv minimum allows 0.5V for regulator dropout.

A likely solution to your problem is to remove or disable the regulator (if that's what it is) and to operate the HIP4081A from an external power supply - typically 12V.

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Added:


  Can I simply desolder [the regulator] from the board and use the naked pads to apply the 12V to IC? 


All care no responsibility advice :-) :
Yes, remove the regulator and feed 9.5 to &lt; 15V to the prior regulator output.  

Do look at the H-Bridge datasheet and convince yourself that it does not "care" if Vhv is say 2V, but I think it is 'uncaring' about this.   

If possible check that the bootstrap gate drive voltage is safe for the FETS - but you'd hope that this was designed. You'd hope.    

Note that the regulator has Vabsmax of 35V (page 5) - I guess that that accounts for the (hopeful) 40V max Vhv rating :-). 

As someone noted, now you have the low VHv sorted (probably)
you can if desired PWM the HBridge to vary the voltage across the whole voltage range with either polarity.
Switch one side high and PWM the other side. Swap sides to swap polarity.