No valve adjustment?

Before going to the hassle of removing the cam, you may want to make use of a dial indicator to check whether the cam lobe at the ticking location is really bad. It could be that there's trash upstream of the lifter (pushrod, rocker, shaft) blocking oil feed to that lifter. How thoroughly did you clean the rockers, shaft, and pushrods before reinstalling?

The "ZOMG, current day oils will ruin your old engine because flat tappets and no zinc!!!" thing is false. It spreads as a result of belief in (and repetition of) misinformation. Yes, one particular extreme-pressure antiwear additive has been reduced in present oils, but it has been replaced by others, and the certification tests for the newest service grades (currently "SN", previously "SM") still include flat-tappet engines with strict limits on allowable wear.

(You might be able to remove the cam with the engine in the car, depending on exactly what car you have and whether removing the radiator and grille provides a clear path. It's going to be a pain in the nuts either way.)