If a new case is absolutely out of the question, you could maybe buy a PCI-E extender ribbon (such as
this one for example) but then you still have the issue of mounting the card somewhere securely inside (or even outside!) the case. Not an easy job.
That thing looks unshielded and cheap. You'd have to deal with the added noise, ringing, overshoot/undershoot and signal skew of the lengthened bus, which might result in reducing the bandwidth between the computer and GPU as the bus retries failed transactions. Worst case scenario: it may even cause cheap hardware to fail. Maybe not though, I don't know how PCIE is laid out electrically, and it may have really good signal conditioning making this entire paragraph one big lie
Plus, the board will vibrate a lot because of its fans and it was designed to use the motherboard's slot for mechanical stability. Vibration eventually breaks things.
I'd cut an GPU-shaped hole on the drive mounting bays, or I'd empty out the case, then take a drill to the rivets to remove the lower part of the disk mounting bay. The rivets and bays are often quite flimsy and easy to pull out, sometimes even without power tools.