It's more expensive than your budget, but if you want something that can handle just about any type of work, I can warmly recommend Fenix HP30R v2.
The Fenix HP30R V2.0 is a professional headlamp with a separate battery case and multiple output modes that is able to produce 3000 lumens.
www.fenixlighting.com
2x removable 21700 cells (included). Battery pack can be carried at the back of the head (it's very well balanced), or carry on a belt with the included clip. Lamp is easily operated even with gloves. Twist clockwise for spot, twist counter-clockwise for flood. Click the button on the twist knob for both spot and flood at the same time. Battery pack also works as a battery bank, and can be charged or supply power over USB-C. The headlamp also support power passthrough, meaning you can remove both batteries (or leave them in) and use an external battery bank to power the light.
Yeah, I have a HP30R. And I have done everything I described above. I love this light. But it unfortunately is expensive.
I primarily got it for the extended runtime with 2 batteries instead of just the usual single cell.
Optionally, I would say Fenix HP25R v2.
The upgraded Fenix HP25R V2.0 rechargeable headlamp is the new standard in high-performance rechargeable headlamps.
www.fenixlighting.com
Whichever headlamp you get, I would recommend anything with the battery stored separately at the back for balance. I own and have tried so many headlamps with everything at the front, and ALL of them have had issues with sag due to being too front heavy.