I've done a lot of recording, and this is a non-issue to me. It's not about the bass or the amp or anything else: it's about how you make it fit in the recording. When I was recording a Carvin LB20, that required a certain amount of manipulation. With my Precision, not so much.
Over the years, I have heard a very few recorded bass lines that have blown me away (as opposed to being workmanlike, journeyman work): on the short list -- excluding absolute masters like Jaco -- are the bass part on Neil Young with Crazy Horse's Everyone Knows This is Nowhere
"Down by the River, and 'Ferry 'Cross the Mersey." A friend and I were jamming on the former last week, and I had to apologize for not replicating what had been captured on tape.
With live playing, "it's there and then it's gone." Do the best you can, and have fun.