So, I was copying those files from C:\Windows\SysWOW64 for the 64-bit build and from C:\Windows\System32 for the 32-bit build. However, it seems that at least on my Windows installation, the 64-bit DLL sits in C:\Windows\System32 and the 32-bit DLL sits in C:\Windows\SysWOW64. The naming of these directories has utterly confused me and resulted in shipping the wrong DLLs version along with Tiled, at least since the introduction of the MSI installer (not sure if it affected the NSIS one)…
Now I realized that WOW64 means “Windows 32-bit on Windows 64-bit”, so the SysWOW64 directory then logically contains the 32-bit DLLs. It does not explain why the folder containing the 64-bit DLLs is called “System32”, but I guess that can be attributed to backwards compatibility and this folder containing both 32-bit and 64-bit DLLs for now.
The fix was trivial now: