Not sure about your wheels as I don’t run disc.
As far as your 1×10 with an mtb cassette I have heard the 9spd mechs work with the new 10spd sti shifters but I have no exprience of this set up (I explored it a bit when I wanted to run a 36 at the back).
My suggestion would be run sram kit and get a RH apex lever as that is compatible with their 10spd road stuff and you can then get a clutch mech which I imagine is something you want running a 1x system. I ended up going sram over bodging shimano stuff as now all my bikes run sram and it makes it easier for carrying spares.
Another bonus of sram stuff is their slogan should be “shifts through shit” because ive ridden it in some horrid conditions and its not missed a beat since it was installed and set up at my LBS.
I wouldnt be comfortable running a bar mounted shifter as you woudl have to get off of the hoods to change gear and you may be riding on something techy at the time.