My suggestion would be to
analyze some bass tracks you like and recognize and understand the elements you like and try to incorporate those elements into what you want to come up with.
When I'm presented with a new tune (whether written by someone else or me) I take a lot of different things into consideration. #1 has to be to give the tune a solid groove and foundation, but beyond that look for little rhythmic or melodic "hooks" that can carry on through the tune and give it an identity. Listen to to the vocal and make sure it's supported properly. If you want to throw some little signature licks in, look for places in the arrangement where it won't get in in the way.
Above all, serve the song. If pedaling whole notes is the right thing to do, do it. You've got your whole career to show everyone what you can do, but if you cut some great tracks being simple, you'll have more opportunities to do that.
