Let's start by reviewing the basics. First, you make the béchamel. Which means you make a béchamel so it is thick enough to coat the back of a spoon, run your finger through it and the swiped area remains. Strain the béchamel to remove any impurities, lumps and whatnot. Then add the cheese. Then use it for your dish.
As the flour in the roux is what typically prevents the milk and sauce from breaking, my thought was that the béchamel is too thin before adding the cheese. So maybe more roux for a slightly thicker béchamel.
Fwiw, I used to make a special cheddar cheese sauce for a dish we served but it kept breaking. We kept the sauce hot in a bain marie during service. After some experimentation and a few questions to our cheese guy, he told me it was because the cheddar we used was too aged. The more aged the cheese, the drier it was. The younger cheddars had more moisture and wouldn't break the sauce. Right or wrong scientifically, we switched to a younger cheddar and the sauce stopped breaking. I never really understood why it worked but it made life a lot easier.