Converting friction shifters to SIS or STI shifters
Since you're going to a lot of work, moving to 8 speed (or 9 or 10) sets you up for the most flexibility.
The main reason is that you need a new rear hub with a cassette-freehub.  There's no good way to convert a freewheel based wheel.  Since you're buying a replacement hub/wheel, its no more work to pick an 8~10 speed one.
The OLD of your bike needs to match the hub, so measure that first.  If its 126mm, you'll have to decide if you want to cold-set your frame.  Is your frame steel? It could be aluminium from the photo, which doesn't cold set.

You will need a full set of parts, plus substitutes for the fittings missing on your frame.
Starting at the top, the brifters should just slide onto your drops and bolt up.  You will need to either carefully unwrap the existing bartape to reuse, or buy some new.
Next are the downtube stops. Your frame has nice but unusual shift levers - the more common ones are two lugs horizontally out the sides.  For them its possible to buy bosses that create stop points.

There are other options - you might choose a &quot;band-on&quot; cable stop like this:

or if you're happy with welding/brazing/2-part epoxy then stick-on ones might suit.

Ideally you'll be able to use the existing cable track under the BB, the photo isn't clear how the internal cable exits the downtube.

If you want to somehow tie the new brifters into the existing access holes on top of the downtube, well that's going to need some engineering and likely a frame builder.
One possibility might be to use Nokon-style cable housing to run from your brifters all the way down to the existing cable entry holes, but that could leave some loops of outer that will look weird.  There could also be stresses on the cable as it passes through at least one 90-degree bend, maybe two.


Additionally, I found that the Klein downtube bosses had some kind of weakness in the past.  Details are from https://www.bikeforums.net/bicycle-mechanics/729717-hillrider-s-klein-stage-comp-fix-shifter-boss-bolt-stripping.html
In summary:  the bosses are weak and there's a refit kit to make them stronger.  However the instructions suggest the existing boss is only bolted to the frame, so it may be possible to unbolt, for a cleaner look.
With the knowledge Top-mount downtube shifter I found this rare Suntour adapter - you might choose to use this as inspiration to make your own, or talk to a machinist to get something made:
   from https://www.bikeforums.net/classic-vintage/1163305-how-use-ergo-sti-shifters-aero-dt-shifter-mount.html some gorgeous photos there
And here's someone who secured normal lugs to his Klein:
 from https://www.bicycles.net.au/forums/viewtopic.php?t=30867

Finally, whatever you do, save the take-off pieces.  Someday you or a future owner might want to restore the bike to original, and these levers are rare.  So please don't throw them out.  You might even choose to sell all the takeoff parts to help fund your changes, but don't bin them.