Quick summary....
I get to the range early and shoot under a covered area.
With the current range setup and early morning sun angle, it results is a darker recorder than I want.
Some neat details get lost or just don't look as good as when recorded with the proper lighting.

As a result, I would like to bring a rechargeable or battery powered light to the range.

As those selfie lights (round ones) any good? Like this.
Or are they more for close-ups?

Do I need something larger like this?

What do you recommend?

What are you lighting up to record? Yourself or your target?

Myself. Figure enough to cover a 3ft x 5ft frame which should be enough to capture shooting a rifle capturing from waist high.

You'll want the bigger one. Get as much light as you can. There's alot of factors that will affect how the video will look (amount of ambient light, distance from light source to subject, quality of the image sensor, etc.)

There are some nice, decent CRI rechargable lights from Neewer or Aputure that would fit the bill nicely.

I’ve seen these in action, they’re amazing:

Click

A "recorder".

Like video?

What you want will depend on the distance of your subject.

Consider a light panel like this GVM 560AS. (You'll need a stand for it, too.) Better options exist the higher you go in budget, but these are what I would consider "budget buys".

I think that ring light in the OP is a non-starter for this purpose. The LED panel in the OP may be bright enough, but I suspect you'll need it within a 24" of you to have enough light and the light will be rather harsh.

This is key. The light falls off very rapidly as the source moves away from the subject (see "inverse square law") and the smaller the source, the "harsher" the light on the subject (i.e., sharper shadows and more contrast on facial features).

I debated about these, but batteries are not included.

Purchasing batteries puts this over $200. Seem like a decent price but not sure if I would get much use of them afterward.

You could get one of those giant batteries to run those off of. Expensive, but you can use the battery for other things. Just search for lithium power station.

According to the description, those lights don't draw a whole bunch of power. Says 29W each. You can get a 250Wh rated battery for less than $200. It'll run both those lights at max power for 3+hrs.