Thank you all for the great suggestions. After careful consideration, I decided to go with a couple of rivet nuts. The trick was finding one with the proper outer diameter. Turns out a 3/8 nut with a 1/4 inner thread fits perfectly. Slides snugly right into place. Give it a good squeeze and there's a threaded hole where the blank hole used to be. Mounted the bumper, threaded the new bolt and it cinched right up without spinning.
I was skeptical at first since the whole structure is required to hold the car as it's pulled up an incline in the event of a tow. However, after doing the installation, I'm confident it'll hold find under those circumstances. Below are a few photos for ya.