Will these Dura Ace components get along with each other?

7402 rear derailleur
SL-BS77 bar end shifters
8-cog Hyperglide freehub

Yes, with the RD-7402 it should shift an 8 speed HG cassette properly, with an extra click on one end.

You can try it with a normal setup method, extra click on the innermost sprocket end, restricted by the L limit screw on the RD.

If you have problems getting into the innermost sprocket, you can try it with the extra click on the outermost (H) end. The amount of cable pull between 'clicks' are not exactly the same in all positions. Again set the RD limit with the screw. If you do happen to go into the 'high' position on the shifter, the cable will go slack.

That combination is actually Shimano replacement spec - 9 speed shifters shift 8 speed 7400 series perfectly. Use the first 8 clicks. The low limit will prevent you from getting to the extra click on the low end.

Awesome...thank you!

I have the DA brifters, but love bar ends...so that's the path I'm taking.

Now, all I need is a set of levers and a pair of bar ends!