I guess your NC outline and the red bike outline are something like this:
The NC state outline isn't a curve, but a filled closed shape which has no stroke. Only the northern border against Virginia is simple, all other parts are full of nodes.
The red bike is a single stroke curve with no fill, it's drawn (or made as Offset) to be the reference how much and where the NC outline should be bent outwards. The white interior of the bike must become totally free, but the new NC outline must be as wide as it originally was and it must not be moved more than the absolute minimum.
I guess only a piece against north-east Tennessee and Virginia border actually need the fix, the southern overlaps can be omitted.
The interior edge of the new NC outline should meet exactly the interior edge of the red bike stroke in the problem areas:
The new midline for the NC outline at the problem areas can be got by making an offset path for the red bike. The offset is (B-R)/2 where B is the width of the NC outline and R is the red stroke width. Unfortunately it's not possible to input (B-R)/2 easily anywhere as a number, it needs SVG code manipulation. I skip it and make the offset path visually.
In my next image the green curve is that offset path. It covers the whole bike, but the extras can be cut off:
How to make the green curve:
- duplicate the red bike, give the same stroke width as the width of the NC outline and green color
- apply Path > Dynamic Offset and move the handle until the offset is visually right. Have a big zoom because visually you can get it right only within plusminus 1 screen pixel.
Close the red bike in the Objects panel (do not delete it, you can make an error and need it). Select the green bike. Apply to it Path > Object to Path to make the green bike free for edits.
Split and remove the unnecessary parts of it with the node tool and Apply Path > Stroke to Path to the remaining part to get a closed shape:
Make two Duplicates for both shapes because boolean operations will eat the other participant. Make an intersection (Path > Intersection) of one pair. It's orange in the next image:
Bring one black shape to top and subtract (Path > Difference) it from a green shape.The result is colored to magenta in the next image:
Bring the last green shape to front and subtract it from the black shape, The result is black in the next image. Select all and apply Path > Break Apart to get the splinters free for removals:
This is what's left after the removals. You can color the parts to black and make a group or combine them with Path > Union.