How can I add "standoffs" to elevate a SMD component above the PCB?
Adding some carefully-cut strips of Kapton tape to protect the vias you're concerned about is an option, but personally I'd just rely on the solder mask for insulation in this sort of prototype situation. You can probe it after the rework to check for shorts, I bet there won't be any.

Additionally, isn't the actual crystal package smaller than the footprint? In your image you've overlaid the same footprint on itself rotated, but the actual component ought to be smaller and might fit entirely within the footprint despite the rotation. Or if it doesn't, perhaps you can find a smaller crystal with the same frequency and similar capacitive loading requirements.