TKS was designed for LEO missions, Soyuz for Cislunar! I believe the designers took advantage of the knowledge the standard Soyuz would not in fact go to the Moon to lighten the return capsule heat shield, thus we'd have to have a modified, heavier Soyuz for return from Lunar space. But it would not be a matter of even another metric ton I would think, maybe half that; also sometimes I have the impression they left all or most of the original design heat shield meant for lunar space returns in place as extra safety margin.
Soyuz is really heavy on orbital maneuvering propellant too, is my impression; again this makes sense when we remember Korolov meant it for return-from-Lunar missions. Note that Americans adopted the higher energy nearly Earth-escape free return orbit as baseline for Apollo; the speed difference at TLI is small between that and a minimum speed Hohmann orbit; the difference has the largest consequence when approaching the Moon. All descriptions of Soviet Lunar missions I have seen assume a Hohmann transfer form of TLI, which is never free return; the spacecraft must perform a thrust maneuver even on a flyby with no attempt to orbit to return the craft to skimming Earth's atmosphere. But then again most of them also have pretty good redundancy in propulsion too as the Moon is approached. If a decision to orbit the Moon is made, I believe a system that places a Soyuz there without using any of its internal propellant will enable the Soyuz to break free and attain TEI of some kind with just that, although that leaves it with very little propellant reserve for midcourse corrections!
Now Luath specifically asked if Energia could boost both a Soyuz spacecraft and its standard upper stage! I suspect it might make more sense to do something quite different, but first of all let us look into that:
Data for Energia from
Norbert Brügge:
Boosters:4 Zenit type (fudging here, I forget how close the boosters are to the separated Zenit!) units, totalling
262.4 tonnes dry, 1228 tonnes of ker-lox, 4 RD-170 engines totalling 29.028 MN sea level thrust implying 31.6276 MN vacuum--I'm going to use Silverbird so it is necessary to know vacuum data; Brügge doesn't give that for SL stages in the Description charts but does for individual engines in the Propulsion charts. Vacuum Isp is 3305 m/sec (Brügge uses effective velocity) or 336.9 sec. These burn concurrently with the core which is 72.56 tonnes dry and holds 703.64 tonnes of hydrogen-oxygen mix propellant. Its engines, four RD-0120, total 5804 kN at SL implying 7844 kN vacuum at 4457 m/sec or 454.3 sec.
Entering all that into
Silverbird launch calculator for a user-defined system with 4 strap-on boosters, to 185 km altitude, 58 degrees inclination (Russians must avoid launching over China!) from Baikonur, I get 126.3 tonnes payload to LEO. Fairings would have to come out of that.
From another page of Brügge's I find that the Soyuz spacecraft of the late 80s through 90s would mass about 7 tonnes, and there is a payload fairing of about 4.5 tonnes removed with the ejection system. Now it is hard to visualize how to install a stack of a Soyuz spacecraft, crewed, under the standard fairing with a second stage attached; or rather we can see how to attach it just fine but when it is ejected, or if God forbid it needs to function as a launch escape system, it awkwardly would be right next to the tank and probably blow it up. They could send it up without the usual LES and uncrewed and send a second Soyuz to dock in orbit I suppose, or design a special fairing. I don't see why the second stage requires any fairing at all except maybe something around its engines to smooth out airflow. I'll arbitrarily recompute with a fairing weight of 8 tonnes discarded after booster ejection, at 130 seconds; that brings payload down to 124.755 tonnes.
The entire Soyuz rocket upper stage used on these missions massed just 25.58 tonnes! Clearly we have over 93 tonnes to spare!
It would be more sensible then to launch the Soyuz and the upper stage directly to the Moon; an escape parabola gives a good ballpark estimate. The sad news is that this gives around 14 tonnes payload more or less.
To my knowledge the Energia system was not designed to shut down the main engines and then restart them, and of course no crewed mission to the Moon is going to launch recklessly into a direct injection to Luna anyway.
To simply place the 32.5 or so tonnes of a Soyuz upper stage plus vehicle stack into LEO Energia is monstrous overkill; the additional 90 odd tonnes would be good for more rocket of some kind to put something into TLI from a parking orbit clearly.
Another approach would be to see how an elliptical orbit might serve us. With a 200 km perigee and 35000 km apogee the mass works out about right. To achieve a Hohmann orbit by burning at perigee to the Moon would seem to require something like a mere 213 m/sec delta V! Assuming the Soyuz upper stage can be burned very briefly to achieve this precisely and then shut down, with a vacuum Isp of 3198 m/sec, we would require just 2.2 tonnes of propellant. I assumed the Soyuz actually masses 8 tonnes instead of 7 to allow for mission enhancements.
Arriving at the Moon, it should be possible to brake down to a low Lunar orbit (circular, not your usual Soviet elliptical one) with a delta V of just 800 m/sec or so. Granting a delta V budget of 2 km/sec for the combined operation of entering and later departing LLO for a return to Earth trajectory, the total propellant depletion from TLI to TEI would be 17 tonnes. The Soyuz upper stage rocket holds almost 23 tonnes, so there would be 6 tonnes left unused! This clearly represents an opportunity to bring some extra tonnage to LLO and leave it there.
It clearly would not be possible to bring along a fully capable lunar lander; the Soviet LK design was already dangerously lightweight for one cosmonaut and extremely limited mission capability. But--for one-way delivery to LLO, I think the system can close with something between 30 and 40 tonnes atop the Soyuz second stage. Thus with two launches, a Soyuz may rendezvous, even carrying some extra tonnage in the form of a mission module--say the propellant for an ascent module on a 30+ tonne Lunar descent vehicle, roughly two or more times the mass of an Apollo LM. Such a vehicle can presumably enable two cosmonauts to stay on the Moon a very long time, perhaps an entire month, while a third cosmonaut remains on station in LLO; then upon completion a small ascent module may return the two with ample Lunar samples to the Soyuz.
I will let others who are bigger fans of the TKS than I am do comparable estimations of it as payload. Certainly TKS was not designed for Lunar missions, though it might be a far better place for a single cosmonaut to remain camped out in LLO for a month! Its three-crew return capsule was not, as far as I know, designed for entry from Lunar return energies however; that would have to be modified. Probably there are more sensible combinations; perhaps the TKS is in the 14 tonne range the Energia could launch on a free return Lunar path; delta V to LLO or landing from such a course is higher due to a faster Lunar encounter speed. If the TKS can reliably do delta-V in the ballpark of 2500-3000 m/sec then it would be a good candidate for a vehicle to be a counterpart of an Apollo CSM, except for the issue of return speed, but that might be dealt with with a modest increase in TPS thickness. Either way any lunar landers would have to be delivered separately; Energia will not launch a full round trip capable package of crewed vehicles in one launch, though two delivers a very handsome margin of performance!