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.
In the past I've done something similar with crystals.
The solution I used for lack of any kapton tape, was to place a big blob of solder on the pads to form essentially the standoffs you describe. Then also tin the pads of the crystal with solder.
Then place the crystal on top of the solder blobs (which have by now cooled solidified) in the correct orientation - it will be sitting a short distance off the board now.
Re-heat the solder blob for one corner to allow the solder to flow and connect the tinned crystal pads to the board. Make sure to hold the crystal steady for the first join so that it sits level.
Now that the crystal is held in place proud of the board, you can reheat the remaining corners in turn to complete the connections.
I wouldn't do this on a production board, but it is easy enough to do for a few test boards.
Ok, this maybe weird but how about soldering some other SMD resistors (very high value) or capacitors(very low value) underneath? It greatly depends on which pads are active and which are passive.