Personally I would not touch the pickups, they are good IMO. I have a Katana 50 and I would say that it's not the greatest amp for metal, for blues no question.
6 things you can do:
1- Play with the height of the pickups (half a turn at a time) and play with the filler screws (again half a turn at a time). If the Jazz is too bright, turn the filler screws clockwise. You way have to raise the pickup to compensate the output level.
2- Get Katana Librarian app for Androids. This changed the way I was dialling my Katana 50. It's way easier with this app than it is with BTS. Start by adding an HPF and a LPF. There's a good chance your Jazz will sound good with just that. Removing some low and some high will also makes the Custom 5 less scooped. Had just enough mids. With the Librarian, it's also very easy to have 4-presets for blues, 4-presets for Def Leppard and another 4-presets for metal. Takes 5 seconds to load them.
3- Get an external close back cab. This will change the response of the Katana 50 and it will sound closer to an amp for metal. I plugged my 4x12" cab on the Katana 50 and it transformed the amp. I also have a custom 1x12" close cab with an Eminence GB128 that I really like for blues, classic rock and hard rock. Don't get me wrong, the Katana 50 is great for the price but a bigger cab and a better speaker transformed the amp.
4- Get a Katana Head and whatever cab to get the sound you are after.
5- Try your guitar on a different amp. Go to a friend or the a music store with that guitar to compare your amp vs another amp. Maybe it's the way you dialled your tone on the Katana.
6- Change your pickups, we are on Seymour Duncan forum after all
Again, an amp and a cab will give you better result IMO because the Custom 5 and the Jazz are already good/great pickups.