Hello,
I'm in the process of designing a patch and I'm trying to get this braided pattern into a symmetrical circle. I've been trying to see if I can attach it to the circle, but I'm unsure how to best go about this. Any suggestions?

I would try this:

Separate the two parts of the 'ribbon', apply one of them as a 'Pattern along Path' LPE along the circle.
Make a duplicate of the result, rotate it a bit and change the fill color.

An alternative solution that does something that can't be done easily with the LPE in its current state as far as I know is to use a two colored group of three objects and the Extension Pattern Along Path found in the Extensions menu in Generate from path. The pattern has to be above the circle it will be put on, and unlike the LPE once it's applied there's no more editing of the original pattern. I just thought you might like this method because your picture was titled "Braid"...
I've attached the SVG so you can see how the object is made, note that the circle can be deleted afterwards but I just unpainted it so all the elements are still there