The travel mode type doesn't really do a whole lot, but if you set it to Walking, it will use a smaller pixel size for your Service Areas and make the polygons a bit more detailed. But it doesn't do anything to affect the actual analysis result.
If your road is "major" but also has a bike lane, you should probably just make it not restricted. Otherwise, bikes won't be able to ride there, which isn't your intention.
Note that crossing major roads can still be a challenge/problem on a bike, so it may not be that inaccurate to show that the area across the street is inaccessible unless you have good infrastructure for crossings and left turns.