I have a 5500K LED continuous video light like this:
https://www.adorama.com/jief150.html?kbid=68009
I use it with a 36 inch octagon softbox.

What kind of filter do I need, to lower the color temp to 3200k?
Can I tape a piece of rosco gel filter between the light head and the softbox? Would it meld from the heat?
Is there something designed specifically for this purpose?

You can search "CTO gel". I believe 3/4 will change 5500K to 3200K. Looking at it's specs will confirm this.

The gel will melt if it touches or gets too close to the LED. You can tape or clips to attach gels to internal diffuser. If you find this a pain then get an Aperture because they have gel holder. You could also buy a variable temperture mono lights. They're not as powerful but its a very useful. Note gels color changes temp with the light's intensity. This means cto 3/4 might be 3200 at 100% and 2800 at 50%

What kind of filter do I need, to lower the color temp to 3200k?

Why not just use the light as it was built for 5500 temperature? Personally, I haven't worked on any shoot that used 3200 (tungsten) lights in about 8 or 9 years. Everyone is moving towards daylight balanced lighting. Daylight LEDs look better on skintones and are easier to mix with natural daylight from windows, etc.

As DJ says why not use daylight LEDs? Though if you have to match a 3200K Tungsten environment you can do this with 3200K or Bi-color 5600/3200K adjustable LEDS.

If using filters a Wratten 85 will take you from 5500K to 3400K. This is what we traditionally used when shooting with film. Watch out for melt temperatures if mounting the gels inside a softbox. Most likely you will have to experiment. See table and Rosco links below :

https://www.timlaytonfineart.com/wratten

https://au.rosco.com/en/color-filters-entertainment-lighting

Chris Young

Don’t forget that conventional CT for tungsten doesn’t work for LED. Lee Zircon does and is quite a different colour to the naked eye. More like an amber? You also lose nearly half your light!