I recently bought a Fuji Touring Series V that has a stock front SunTour MounTech derailleur and a non-stock Shimano Deer Head rear derailleur. I recently threw a pair of 700c Shimano RSX hubs and Mavic Rims. I am entertaining the idea of upgrading the rear cassette to a larger gear range as well as a pair of MicroShift bar end shifters. I just don’t know how compatible all this is and what I can use and what I can’t use.

The FD is fine as the bar ends should be friction on the left. You'll need an indexing RD and shifters matched to the cassette. Sticking with friction is easy peasy.

Yeah. The RD needs to be Shimano SIS compatible in order to have index shifting.

That looks like a Suntour Shimano* 6-speed freewheel, not a freehub-type cassette. With some possible respacing of the rear axle, you could get a 7-speed freewheel on the back, but not any more than 7 speeds without replacing the entire hub / wheel (even if you source an 8-speed freewheel, the rear axle is likely too narrow to accommodate without running out of threads, plus the unsupported axle length would probably result in breakage). If the Microshift bar-ends are intended for 8 speeds or more, you're gonna need a completely different cassette-hub wheel or rebuilding your existing wheel with a cassette hub (not for the faint of skill).

If the Microshift bar-end shifters are indexing-only on the rear, you will need a different rear derailleur, as I recall the Deore deer head rear is friction-only (and mildly collectible if it is removed). Not surprised it has a mismatched rear derailleur - the OEM Suntour Mountech rear derailleur was a design too far in terms of innovation - placing the spring inside the pulley resulted in crud intrusion and pulley seizing.

*missed the VIA stamping on the first glance

That TSV is hot.

If you're using friction- Suntour/Shimano/Microshift is irrelevant. For the most part.

If you're indexing, the shifter, derailleur and the cog spacing needs to match.

Love the Sugno Aero Tour crank, though.

Any chance of all of us talking the OP into keeping the current half-step gearing with a 6-speed indexed rear and a nice modern ramped freewheel?

The OP did say he has a new wheel with an RSX hub, which is most likely a 7 speed HG freehub with 126mm spacing

Love the crankset.
If it were me, I’d sell the Deerhead to someone who wants it and replace it with literally any newer Deore RD from MT60 onwards.
Would replace shifter cable as well with modern drawn stainless Shimano (we used to call that older DA/Deore stuff ‘rope’).
Result will be vastly superior and reliable shift quality.
With a $6 Amazon dropout extender you can shift a 36t big cog…