Good news, the KVTK should be powerful enough to put the Soyuz in lunar orbit.
That "cryogenic" ISP really helps !
Dv = 4542.03 * ln(31500/11650)
Dv = 4517.87
Now, if we add a Fregat atop the KVTK (heavy payload for the "poor little" ProtonM
)
I assume a LM of 10 tons for now
Dv1 = 4542.03 * ln(24000+6400+10000/4400+6400+10000)
Dv1 = 4542.03 * ln(40400/20800)
Dv1 = 3015.34 m/s ; velocity gain at the end of the KVTK burn
Dv2 = 3256.92 * ln(6400+10000/11050)
Dv2 = 1285.99 m/s ; additional gain of velocity after the Fregat burn.
DvT = Dv1 + Dv2
DvT = 3015.34 + 1285.99
DvT = 4301.33 m/s ; total velocity gain of our 10-tons payload after the burns.
So, on the paper a 10-tons LM is go for the Lunar Orbit ! But we have to consider that the KVTK will need to perform a burn for LEO insertion (because the KVTK+Fregat stack is well over the ProtonM payload capacity).
If we are really too short on Dv, there's an alternative that need another launch :
Launch 1 : Send a LM in LLO that has just enough propellant for the final landing phase and the ascent phase.
Launch 2 : Send a Fregat in LLO.
Launch 3 : Send a "Fregat2" in LLO
Launch 4 : Send the Soyuz in LLO, then :
a) The Soyuz docks the LM and power it up (fuel cells plugged or switched on).
b) Crew transfer to man the LM.
c) The manned LM performs a rendez-vous & docking with the orbiting Fregat.
d) The Fregat brakes the LM until the final landing phase.
e) While in free fall, the LM undocks and perform an avoidance manoeuver.
f) The LM lits its engine. Final landing phase : the pilot has to show its skills !
g) Lunar Landing and EVA.
h) Liftoff ! The legs and the chassis that supports them stay on the Moon.
i) The LM achieves LLO (no more than 30-40 kms of altitude are needed).
j) The LM burns its last drops of fuel to perform a rendez-vous & docking with the Soyuz.
k) Undocking.
l) The Soyuz performs a rendez-vous with Fregat2.
m) TEI
n) MCC
o) Reentry
p) Landing & recovery (at least !)
Edit : it could be well advised to have another rocket (I guess that Baikonur pads will all be used in the operation so probably elsewhere) ready to launch another Fregat in LLO, just in case the Fregat2 fails and the TMA is stranded in LLO... The expedition could carry food & water for 1 month, just in case...