Your AP freewheel (7 speed) should work well with the X-1 rear derailleur. You should use compatible 7 speed accushift levers. On my touring bike I have a Winner Pro 7 speed touring ratio freewheel, an X-1 rear derailleur (I bought this NOS from Yellow Jersey for $55), and Superbe Pro index down tube braze-on levers. The front shifter is friction to an older Specialized road triple crank paired with a SunTour XCD rear derailleur. I’d have to look to see what chain is on there (I have multiple road bikes and I haven’t ridden this one in several years.
The shifting for this 3 x 7 combo is very good. Excellent actually. Different than Shimano I think since the clicks are a bit more positive (less soft) compared with Shimano.
Keep in mind that the AP freewheel has different spacing on the smaller cogs compared to the larger cogs. Sorry, the exact specification doesn’t come readily to mind, maybe the archived PDF file from Yellow Jersey lists it. I want to say that the bottom 3 cogs are like 2.8mm spacing, the remaining ones are 3.0mm spacing. It has to do with the sweep of the rear derailleur as it responds to the cable pull of the shift lever.
To the person who mixes and matches Shimano with the SunTour, I would say more power to you but I believe best performance comes from matching parts. People that get these mongrel systems to index sufficiently I believe are relying on Shimano’s patented floating upper pulley “Centeron” in order to approximate proper shift spacing. Better to use all matching parts, I think.
I also run a he SunTour Accushift 8 speed Superbe Pro cassette hub with command shifters in a 3 x 8 configuration which I cobbled together custom and it is equally awesome. Since 8 speed accushift cassettes that are not “Microshift” (with the outer 2 cogs threaded together as a sub-assembly) are about as rare as a white Unicorn, I had to merge an extra cog and spacer from a spare 7 speed AP-II cassette. It turned out fine and uses the same tighter cog spacing on the smaller cogs compared to the bigger cogs.
The quality level of the Superbe Pro cassette hub is comparable to Dura Ace 7400. Very heavy but lifetime durable. My freehub body got gummy and the chain would go slack when coasting so I drizzled a few drops of Pedro’s Synth Oil in there and it freed up almost right away and all is well now.
If you are unable to source the Command shifters and only able to locate the down tube levers but desire a handlebar mount there are still options. While I’m not sure about putting them onto bar end shifter pods (I think SRAM might make a set), Gevenalle sells there Audax brake levers that have braze-on mounts. These are not cheap @ $160, but represent a decent value since they are high quality.
Let it us know how you move forward with your project and if we can be of more assistance.