WT, the only other possibilities I can think of-and they are unproven and possibly reckless to use on the worlds greatest oil field are some type of surficants to cut the immobile oil out of the reservoir rock or possibly microbiological enhancement, injecting bugs that will lower the viscosity of the crude.
But surely Allah isn't going to desert us now!

I wonder, if the source rock is some sort of carbonate, could you collapse the pores by flooding with some acid solution, maybe followed by a shock wave? The oil would be wrung out like water from a sponge.

The reservoir rock in Ghawar is a carbonate. A dose of acid serves to open up new paths for fluids [both water and oil] to the well bore. This is usually useful only when the permiability of the reservoir rock is low. If you only have the very top of the formation open either through a conventional completion, a horizontal bottle brush smart well, or whatever -- opening up the formation probably means opening it up vertically as well as horizontally. The result would in most instances be even more water.

In a test tube, you could disolve a chunk of the reservoir rock with acid, but you would also do bad things to the immobile contained hydrocarbons ... and even if the acid wouldn't digest the oil this approach would be completely uneconomic in a real world setting as the reservoir by volume is probably in excess of 85% rock; the contained water dilutes acid; a lot of the fluid you obtained would be water; and [if your theory actuall worked] "collapsed" pores would equate to zero permiablity / porosity in the vacinity of the well bore.

If you really wanted to try to improve the oil cut you could try a "polymer job", but that approach wouldn't get you a thousand barrel a day well water free well by any stretch of the imagination. BTW, my limited experience with polymer treatments has not been good, but they are in certain instances very useful in extending the economic life of a well.