Trying to wire 240 disconnect panel from 3 wire from main panel to 3 wire spa (2 hots and ground on both sides)
When hooking up hot tubs, don't fool around

It can hurt people.  It can also ruin hot tubs. 

The lack of neutral at the hot tub box might be a problem

Your application doesn't need neutral at all.  (According to what you've said). 

However, a couple weeks ago, we had a question about using American GFCI breakers in the Philippines, where some houses are wired exactly like your circuit there: two hots, no neutral, and a ground, theoretically in the middle between the two hots.   It turns out that some GFCI breakers actually need neutral on their input side - to power the GFCI mechanism itself.  

It would not be legal in the US to "bootleg" that input neutral wire to the ground.  Bootlegging also wreaks havoc on any GFCIs that would be upstream, not that you would need any. That may be why TPE is asking about your panel breaker.  In his case, we counseled him to install a tiny autotransformer to create neutral from the two hots.  It worked. 

Honestly, if it were me, I would take back that GFCI spa panel you bought, buy the correct Siemens GFCI 2-pole breaker for your main panel, and then get a simple/cheap non-GFCI subpanel or shutoff switch for this location.  That keeps the GFCI in the dry, warm inside, where it will last a lot longer. 

But if you can't do that, you could try hooking it up without any neutral connected on the input side.  Maybe the GFCI will function correctly.  It is a spa panel after all, having no neutral on the input side is perfectly foreseeable since spas don't need neutral.  If it still doesn't function correctly, I would go talk to your AHJ, and see if they'll sign off on bootlegging netural just for that GFCI device.