First off, I moved your post out of characins and into the general Tropical and Freshwater Fish section. While many stores do list White Cloud Mountain Minnow as a "tetra" they are not, they are a cyprinid. But considering the reason for your question, as other fish will likely be named, I moved the thread into the general category rather than cyprinids.
Now to your question. This is not a good fish with Betta, for a couple of reasons. First, temperature. WCMM require cooler temperatures than many of the "tropicals" and room temperature is as high as it should ever be. Betta by contrast are true tropicals, and warmth is best. Aside from this, WCMM are active fish, swimming-wise, and this is not what you want to put in with sedate fish like Betta. You will notice the fish name shaded, so you can click that for the profile with more detail.
I am with the majority of aquarists who consider Betta as non-community fish and they are best on their own. Some substrate fish can work. The issue here is two-fold. Your Betta may be docile, or may seem so now, but combine him with some species of fish and I can promise this would end. But even moreso, the Betta is a likely target for many of the small shoaling fish like tetra. With a Betta around, these otherwise peaceful fish can turn aggressive and nip the Betta's fins which will severely stress out the Betta. A 120g is a fairly small space to a shoaling fish, and this often makes it even worse.
I will assume your Betta is content in his 10g home, and he might well be happy to remain on his own.