Owning a couple dedicated bicycle lights in the $40-$50 range, I’d say go with a decent flashlight with a bicycle mount- because those ones that use the typical 4x18650 style battery packs usually have runtimes in the 1-2 hour range and 5 times as long to charge- that and the long cords are a pain to manage if you’re not doing a permanent install.
A well regulated light can run for that long on a single 18650 battery.
Decent runtimes are probably going to have you stick in the 18650 style light. And some of those holders require the light to have a clip. I have an Xtar B20, which I reviewed with run time tests.
and you can get it for $37 shipped right now.
My only complaint would be that the light is a bit bulky- so get a good mount.
Other than that, maybe something from Fasttechs Convoy line. Most can be had for around $20. I haven’t tested runtimes on them though, and many of the smaller convoys require unprotected batteries. Anyway that’s my two cents, from lights I have personally tested. The other option would be a magnetic control ring light but those are more than $50 for a light.