I like my VS4 passive, it's a great instrument, But I love my Sandberg MarloweDK.
What you are describing are the 2 inherent qualities of 2 very good basses.
Depending on what you play and who you play with, and what they are playing it's only you can decide what you want.
Your strings and cab are great, your head is good too, although not my favourite for powering your sound, but if you like it, don't worry.
I would do nothing, play the Jazz live mainly, have the VS4 as a back up, or you may find if doing any recording or certain types of songs it feels a bit better.
Don't knock pedals, some are really good and you don't have to spend the earth on them. A really good cheap one that I know plays well with the VS4 is a Hartke VXL, you can get them for £40 second hand and it has a DI out.
You have quality kit, enjoy them for what they are, but definitely do not put a hole in the VS4, worse case, sell it on.