Two ways:
Disable the Refresh Mods Automatically, then copy the new DLL and files directly into TempMods. Then use the Mod Launcher as normal.
or
Copy the files into the environment, after running Starvation Mod once through the Launcher, then manually copy the files in as you've been doing. From then on, just launch 7daystodie.exe directly, without using the Mod Launcher.
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It's client-side only, however, some Mods do package in their server files, which can be found under the TempMods folder, if they do.
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INFO 1 _7D2DLauncher.frmMain - Copying D:\Games\7D2D\My_Mods\SMX_BigBag_plus_Lawnmower\TempMods\7DaysToDieServer_Data\Managed\Assembly-CSharp.dll to D:\Games\7D2D\My_Mods\SMX_BigBag_plus_Lawnmower\7DaysToDie_Data\Managed\Assembly-CSharp.dll
Looks like things were working, but then copied over the Server's DLL into your game folder. Try to exclude it from your zip file, and see if it works better for you.