Ahem... It's the anandamide receptor (CB1) and anandamide is arachidonic acid ethanolamide. But you are right, the possible binding orientation in the receptor is U-like, with the end of the unsaturated C20-fatty acid tail coming close to the ethanolamide head.
There are several studies about aminated THC analogs, but they didn't yield something interesting, simply because the receptor doesn't need a protonated amine for proper binding (like the serotonin receptors).
The most potent propyl-sidechain modifications are the 1,1-dimethyl-octyl and the 1,2-dimethyl-heptyl variants (which are both incredibly long lasting).
And then there is the yet secret superpotent CB1 agonist "mahanandamide"...