I think you are saying that, when the chain is in the smallest cog, to shift to a larger cog you have to apply more force than you'd like to the bar end lever.

I would leave the bar end lever in the position for the smallest cog (shift lever pointing toward the ground), and try shifting by using your fingers to pull the cable away from the downtube. See if the derailleur and cables work okay, without the shifter being involved. The RD should start moving and the chain derailing, as soon as you start pulling the cable, if not then adjust out any slack using the barrel adjuster, limit screw, and if necessary the cable clamp bolt. Also check that the RD cage is vertical (viewed from the rear), the pulleys don't have too much side to side play, and that the upper pulley is not too far below the cogs.

Then shift to the largest cog, by pulling the cable with your fingers, then stop turning the pedals. The cable will now be slack. Push and pull the cable to make sure it slides freely in the housing and ferrules. Also move the bar end shifter through its range to see if there is anything binding up.

Then remove the rear wheel and, while pushing the RD inward with your fingers and letting go, see how much you have to tighten the bar end shifter to hold the RD in position against the RD's spring. You might use a drop of blue loctite if you're worried about the screw coming loose on the road.

Assuming all of that is working normally, then I'd say the force required for the shift "is what it is". I can't think of anything else, anyway.