Should I ignore Google’s rendering issues?
I'd take them seriously as much as I can.
What your question tells me is that some bots are capable of rendering javascript and some are not.
... However, the site is getting indexed as unique content to our site is easily searchable.
Then the bot that indexes pages must have the ability to understand Javascript.
I have noticed that the cached versions of our pages is just the pre-js splash screen.
I'm guessing your website is Javascript heavy. You're in luck for having the pages indexed. You need to have text on the page that does not rely on Javascript then all Google bots will process the page much better.
Also, on http://www.webpagesthatsuck.com/does-my-web-site-suck/does-my-web-site-suck-checklist-part-one.html the author mentions in the checklists for bad website design:
Our site breaks when visited with the Javascript turned off. (People turn Javascript off for security reasons.)
I'd suggest going over the entire checklist, making sure your site doesn't score any check marks in that list then test it again with the google bots.