Good to know the Lollar works well with EQ to let you sculpt a tone you want, as opposed to need EQ to remove something you don't want.The key word for the Lollar is NATURAL.
To me, it doesn’t sound like it’s trying to be too vintage or too hi-fi. It doesn’t seem to be marketed to a specific genre. It just reports the natural sound of the bass.
It gives you such a quality signal that you can push it through some radical EQ, or leave it totally flat, and the whole range still sounds good.
For example:
Our FOH engineer is sometimes looking to boost some high end so that the bass can remain articulate in the mix. This would usually bring out some strange honk in my fender pickups. But with the Lollar he can boost whatever he needs, and the bass still sounds natural.
Yesterday I tried lowering the Player pickup under the G, LPF a little above 5k, and amp EQ small cut at 4k (shelving), to remove some highs or upper mids - to me they sound more like a whine and less like a honk - I wonder if it is similar to what you had with your Player pickup.