My left hand is too far down to help, so I can't use it. How would I play it?
I would play the C an octave higher. Yes, you can use the pedal or even just play the low D staccato to free your left hand to help, but it's still a big and awkward leap. At a fast tempo it will be very difficult to execute correctly.
Raising the C by an octave is no crime in music of this style. On top of this, the composer or arranger should have made some provision for players who can't span a tenth, and the fact that there is no such provision means that most players will have to make some compromise or other. The most natural such compromise is to revoice the chord in the right hand.
If you find the change in the chord's sonority to be excessive then your only other options are to release the bass D (with or without the pedal) or to use your nose.