I faced the exact scenario described in the opening post last week-end. I was on a delayed harvest river and hooked a beautiful rainbow (~15in). I could not see the fly until it opened it's mouth since it was hooked quite deep. I tried to remove the fly but could not reach it w/o inflicting further damage. I tie my flies with barbless hooks so not sure why the hook was so deeply engorged. All the while the poor trout was bleeding..
Made me sick to my stomach but there was nothing I could do except revive it the best I could and watch it slowly swim off.