If your PA mixer is anything like the Behringer, it will have far superior eq control than any preamp/DI box you’re thinking of buying.
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This. The Behringer is the budget version of the Midas so it should be good to go. The advantage to this is you can tweak your "ideal" bass tone on your preamp (or HX Stomp in my opinion) and take your AUX feed pre-EQ/faders off the board and always have your same tone in your ears. Then FOH can use the powerful parametric EQ to tweak your FOH sound for the room.
Edit: having made my second recommendation for the HX Stomp I reread your original post and see it is about double your budget, so sorry bout that. in my opinion everything you get for the extra is well worth it, but its still a lot more than what you said.