"The solution may be obvious to a regular user of POL, but I'm stymied. Any feedback would be appreciated. Thanks in advance!"
What do you mean with that? I'm not clear on that part. As far as I know, e-sword.net are same apps with sabda.org, and it just need installer on wine, no need for dependencies, just download the exe/msi, and install it normally, and it will works as it's.
Probably if you need more ensuring the need, you can see the winetricks/dependencies it need as listed :
• POL_Install_corefonts
• POL_Install_d3dx9
• POL_Install_tahoma
• POL_Install_msxml3
• POL_Install_msxml6
See https://www.playonlinux.com/en/app-2606-eSword.html#expander_description_12542
PlayOnLinux for rest of life. Thanks for the maintainer!