Thanks for the helpful replies everyone.
The oval washers appear to be the best solution depending on how wide they are. They have to be able to pass through the holes in the outer rim (the bit the rim tape sits on) and those holes have a diameter of 8.5mm.
@Ajax Bay where did you get the 9mm measurement for the oval washers? I can't find the dimensions listed anywhere. I managed to find the old eyeleted rim and the spoke bed appears to be a slightly different profile, though it's hard to tell for sure. The new rim seems to have a more evenly curved spoke bed, rather than flat with an angle to the side wall. Also, why would the ERD change from the previous wheel if washers are used? Surely the lack of eyelets would reduce the ERD, thereby cancelling out any increase from the washers?