I understand the OP - I have a similar problem due to the fact that our Open Sic repertoire is quite similar:
Velimirovic, Yugoslav Attack, some Maroczy Binds, ML Sveshnikov and what about the Najdorf?!? Playing an anti-line is one option (e.g. 5. Bd3 or Qxd4 lines), but let's first think of the issue within ML boundaries:
As a Najdorf player myself I am not convinced at all with tries like 6. h3, g3, a4, f4, Rg1 (the 'freak attack' mentioned already), Qf3 (a rather fresh try). Also the more established 6. Bd3, Be2, Bc4 don't pose me great problems. It's only 6. Be3 and especially 6. Bg5 that I consider critical.
On the other hand,
consistency with the OP's chosen repertoire is very, very important here, e.g. 6. g3 Nc6 or 6. g3 g6 and you are out of your repertoire.
So I think the 2 most consistent choices available are:
a) 6. Bc4 e6 => I am not satisfied with it as White (esp. vs. 7. Bb3 Nbd7), nevertheless it is the most consistent and you only have to face 6. ... e6 setups. Play is very tactical which is a minus considering what the OP requested (but Velimirovic is very tactical also!?!?!).
b) 6. Be3 e6 or unfortunately 6. Be3 e5 => so you face 2 setups here (a minor concern is to learn sth vs. 6. ... Ng4 7. Bc1 Nf6 8. f3 Qb6). Though it is the ML, it is more about concepts in comparison to the 6. Bc4 lines (see also TN's comment above). It is also more consistent with the Velimirovic & Yugoslav due to the possibility to go O-O-O (which is difficult with 6. Bc4).
If you decide between one of the above, then you can also consider what to play vs. Schevy setups, e.g.:
a) 6. Bc4 e6 7. Bb3 Be7?! 8. g4 is a nice option and is even more obvious if you also use the Keres Attack vs. the Scheveningen (btw if you do play the Keres, you should seriously consider 6. h3 vs. the Najdorf - only problem imo is the Dragon reply).
b) 6. Be3 can also be applied vs. the Scheveningen (if you don't plan to play the Keres) and is also good vs. the Taimanov.
my 2c