Is there any way ... to get the routed edge in these tighter areas?
Not with that router bit. The bearing size determines the minimum clearance at an inside corner. You can have a full roundover if there are rounded corners or have sharp corners that the roundover misses, it's an either/or.
Would a chisel be appropriate here?
Yes, chisels and gouges to get into tight spots.
Your chisels must be very sharp for this to work well. Sharper edges are good for all woods but are especially important when working softwoods as the pale earlywood is particularly soft or spongy and is hard to cut cleanly using edged tools (paradoxically a harder wood cuts better, although there is greater resistance).
If you need tips on how to sharpen chisels, see these previous Q&As:
Is there a 'best' way to sharpen an edged tool like a chisel?
How do I sharpen curved tools like gouges?
What criteria would want me to bevel my chisel in a certain way
And some help in determining sharpness:
How can I tell if wood turning (lathe) chisels are sharp?
perhaps another technique
This won't work if you want the letters to overlap as you have here but if it's acceptable to have them just touching there is a trick, and that is to saw out each letter separately and round them over individually, then glue them to a thin backer board to bring the word together with the letters touching.
Source: Making Wood Signs by Patrick Spielman