Continuation of the tweaking and hacking... to no avail.
I removed the "have an engineer on board" condition from the save file, but the part was still impossible to repair. I hacked the file a bit more to have the mission request to repair the opposite panel, in case there was something wrong with the parameters of the original one. This time, the "original panel" was clickable and hoverable, but the opposite panel, the one to repair was his turn unclickable and unselectable, exactly like the other one was. So it seems that there is an issue with the "RepairPart" parameter.
In the end, I hacked the save file once more to remove all parameters from the contract and replace them with a "Keep stability" parameter, copied from another contract. (Before that, I tried to copy other kind of parameters, such as the tourist contract I was trying to cover in the same mission, but this did not trigger the completion as expected.) Using this hack, when loading the save and after the 10 seconds of keeping a stable orbit, the contract was finally declared as done. So, not a real solution... It's a pity because these repair contracts are fun (RDZV + EVA).
BTW, this is the full list of my mods, but honestly I don't see which one could be interacting here. ReStock maybe? Or ClickThrough Blocker?
BreakingGround-DLC (BreakingGround-DLC (unmanaged))
Chatterer (Chatterer 0.9.99)
Chatterer Extended (ChattererExtended 0.6.2)
ClickThrough Blocker (ClickThroughBlocker 1:0.1.10.15)
CollisionFX-ReUpdated (CollisionFXReUpdated 1.1.0)
Crowd Sourced Science (CrowdSourcedScience v6.0)
Distant Object Enhancement Continued (DistantObject v2.0.2.0)
Distant Object Enhancement Continued default config (DistantObject-default v2.0.2.0)
Docking Port Alignment Indicator (DockingPortAlignmentIndicator 6.9.2.2)
Kerbal Alarm Clock (KerbalAlarmClock v3.13.0.0)
Kerbal Engineer Redux (KerbalEngineerRedux 1.1.8.3)
Making History (MakingHistory-DLC 1.11.0)
Module Manager (ModuleManager 4.1.4)
PlanetShine (PlanetShine 0.2.6.3)
PlanetShine - Default configuration (PlanetShine-Config-Default 0.2.6.3)
RCS Build Aid (RCSBuildAid v1.0.5)
Real Plume (RealPlume 2:v13.3.2)
Real Plume - Stock Configs (RealPlume-StockConfigs v4.0.5)
Reentry Particle Effect Renewed (ReentryParticleEffect 1.9.1.1)
ReStock (ReStock 1.1.2)
scatterer (Scatterer 3:v0.0723)
Scatterer Default Config (Scatterer-config 3:v0.0723)
Scatterer Sunflare (Scatterer-sunflare 3:v0.0723)
ScienceAlert ReAlerted (ScienceAlert 1.9.8.9)
SmokeScreen - Extended FX Plugin (SmokeScreen 2.8.14.0)
Spectra (Spectra v1.3.3)
TextureReplacer (TextureReplacer v4.3.1)
Toolbar Controller (ToolbarController 1:0.1.9.4)
Zero MiniAVC (ZeroMiniAVC 1:1.1.0.1)