Best solution for 2x8 index shifting
Better 8 speed shifters than exist now are pretty unlikely to start existing in the future, so any recommendation is unlikely to become obsolete.
For robustness, the hot setup is SL-BS64 shifters (the ones you already have if they're Shimano) on Paul Thumbie mounts. Loctite the heck out of the mounting screw for the shifter part. They perform great too, provided you enjoy using thumbshifters.
If you really want trigger shifters, another avenue that would produce good results is any higher end flatbar road set like the one you mention, or any left friction thumbshifter plus the best 8-speed Shimano right shifter you can find in good condition. XT, LX, and Deore have all existed, and most will be better than your Claris. (I will say that the tactile feel of the Claris set you have is of course mediocre, but they work fine, so any serious performance problems you're having with it make me suspect other problems, i.e. cable/housing friction or a slopped out RD.)
Any pre-11-speed Shimano flatbar road left shifter should work in conjunction with any 8-speed double Shimano road front derailleur. There are other things that could be causing your bad front shifting, ie weird chainring spacing or bad FD positioning. What won't work well is putting a mountain shifter on a Shimano road FD. They pull more cable. One can usually make it work by starting with a bunch of cable slack, but it's not pretty.