IMHO Mission module would be integrated with Block-DM and will have a passive RDS unit and there is no SEPARATE Mission module.
There is a delta-v problem and various prohibitive technical problems for this plan.
Multiple variants of Block-DM are currently in use. I pick the one used in Proton:
Block DM-2 11S861
Gross Mass: 17.3mt; Empty Mass: 2.3mt; isp: 352sec; Lox/kerosene
If you max-out Proton you end up with a 4mt mission module + RCS attached to the Block DM, which is quite alright for our purposes (3 crew in Soyuz for an 8 day lunar swingby mission). Now, let's forget for a moment that the Block DM's loiter time on orbit is very limited and a very quick rendezvous between our modified Soyuz (let's call it Soyuz-TMA-X) and this Block DM-2+MM stack is necessary for mission success.
Even if we assume 7.2mt for this modified Soyuz-TMA-X (it probably will weight more due to heatshield changes of the capsule), the Block DM will not be able to provide the required delta-v for TLI. Not even close. Even without a new mission module, the Block DM might only just barely provide the required delta-v for TLI for our 7.2mt Soyuz, but without margins necessary for a safe mission architecture.
All that said, using a mission module with the current in-use Block DM on a Proton for a Soyuz lunar flyby is impossible. A new larger version of the Block-DM with better loiter time, RCS etc. would have to be developed. And if you want to take a 2-4mt MM up there with it, Proton doesn't have the capacity to do so, you need a larger rocket (Angara is still too far away to baseline it for this job...).
I see a different variant of a near-term lunar swingby mission much more likely, one that can be accomplished more easily and with enough margin to allow for a safe architecture. A dedicated modified Soyuz for only a crew of two, involving modifications to the Orbital module and the capsule and the use of Briz-M as the EDS (Briz-M: 22.1mt gross, 19.8mt propellant, isp: 326sec, N2O4/UDMH). NO dedicated mission module at all, the reduction in crew size and modifications to the orbital module (toilet etc.) are enough for an extended space flight. While Briz-M has lower isp than Block-DM, it has a maximum loiter time of 24-30 hours which allows for safe on-orbit rendezvous and is sized appropriately (22mt) to fit on Proton as a payload while still providing enough delta-v for a slightly heavier Soyuz (Soyuz-TMA-X of maybe 7.5mt instead of 7.2mt) for a lunar swingby of a crew of 2. The Briz-M of course needs modifications to its RCS system and a docking mechanism.