technology recommendation for LAN Dashboard
Both of these technologies are certainly mature enough to run Mission Critical applications, you just need to look at the number of big sites already on the internet that are running these technologies, so from that point of view you shouldn't be concerned.
You only need to worry about your learning curve, if you feel confident in learning them well enough to write quality code for your client then go for it. Have a look at each of them, decide which technology you would prefer and get coding.
Hope your application goes well :)
I'm about to start a fairly large project for a mid-sized business
with a lot of integration with other systems (POS, accounting,
website, inventory, purchasing, etc.) The purpose of the system is to
try to reduce current data siloing and give employees role-based
access to the specific data entry and reports they need, as well as to
replace some manual and redundant business processes. The system needs
to be cross-platform (Windows/Linux), open source and is primarily for
LAN use.
My experience is mostly PHP/web/app development, but I have developed
a few LAN apps using Java/Servoy (like Filemaker).  I found Servoy to be very rapid and to easily make use of different data providers (DB products), but it's not open source, and any non-standard development is in Java/Swing (which is verbose and takes a lot of time).  
I'm interested in learning Python/Django or Ruby/Rails - but I'm not sure if these are the best solutions for building a mission critical data entry/reporting LAN app.  Is a web client/server really a good choice for this type of application?  
Thanks in advance for any tips/ advice.