Hi Stephen - that is indeed very strange. Here are some things I can think that might cause this issue:
1.) Embedded Font issue: Is the text embedded, or is the font the text is rendered in being embedded? Check to make sure - weird things can happen when fonts are embedded in one swf and loaded into another. Try un-embedding the font and see if that fixes things. If that fixes the issue, you might need to embed it in both swfs to make it work, or try various combination of embedding in one or the other.
2.) Some weird CSS styles: are you styling the text using CSS? If so, make sure the styling is being set correctly, or that styles in your main app aren't clobbering the styles used by your RSL.
3.) The actual text data: put a few trace statements (or alert windows) into your RSL and make sure the text is actually being set. It's possible that those labels aren't correctly receiving their data. I realize this is somewhat automated in Charting, but some of the axes require label data from dataproviders and I thought you might be doing that.
4.) The axes: make sure the axes are still being set correctly in your RSL, and that nothing in your main app is clobbering the type of axes being set on the chart.
I hope this helps get you started - in general I'd just fire up ye ol' debugger and start working backwards to figure out at what point the text disappears. This will help you narrow the issue down to something more fixable.