Looking for a remote that will reliably shoot as if the button is held down

You can get a wired remote on eBay for $£€ 10

to which you can add extension cables for about the same price.

You don't even need to put batteries in for a simple shutter release, only for the timer functions. As all it's doing is closing a contact, you can run quite a long way. They have a 2-stage press, same as the camera's own release & a lock to keep on shooting. The last part of the cable comes in many flavours for different camera types.

I'm not sure about the Panasonic camera, but with the Nikon D7500, you don't need a remote that holds the shutter for a long time. You can do single-press shutter button exposure bracketing by setting the bracket function and setting the interval timer function.

First, as you're probably doing, set the number and EV range of your brackets by pressing the BKT button.

Secondly, set the interval timer to start Now, taking 1 interval of as many shots as you set in the Bracketing function. You can also set a short delay between shots (I usually keep that delay to a minimum). By doing this, with a single press of the shutter button (or the remote's button), you will take, say, 3 bracketed shots automatically. The next time you press the shutter button, you will take 3 more bracketed shots.

When you're done with bracketed shooting, you have to remember to turn off both bracketing and the interval timer. I know, it's a bit fidgety, and easy to forget to set (or disable) both modes. But it's how it's done with Nikon DSLRs.

With both your Nikon D7500 and the Panasonic FZ300 bracketing can occur one exposure per shutter release press, and any remote will work for that. With the Nikon you just use the basic BKT button/mode (and step settings) **and the shutter release set to single shot mode. And with the Panasonic you set it to "Single" in the Auto Bracket menu.

(**unless you are really good at controlling the number of shots in continuous mode)