You shouldn't jump into an extreme cutting phase if you want to prevent muscle loss. But that really depends on a person's goals.
It can take some time for your body to become used to the new muscle, but I cannot see waiting months before cutting. For you I would take just 2 weeks to help adapt, then slowly reduce the calories and add in some HIIT and possibly (should I use the word) cardio until you have about a pound of fat loss a week.
You probably have a body type that will lose that fat fairly easily, so your goal is to just make sure it is mostly fat you are losing.
It helps to have a food log, or at least a planned diet in place. It is too easy to have calories bounce around if you are not paying attention. But as I stated above, start slowly cutting out calories. This could all come from carbohydrate sources, or a mix of carbs and fat. This cut could be as simple as subtracting 100 - 200 calories from your daily intake every 1 to 2 weeks until you hit that 1 pound a week point, and keep an eye on your progress.
If your weight is going up, it could be going up faster then your increase in strength on that exercise. Chins have always been one of my week points. There are plenty of articles here on chins and pull-ups.
Really you want to gain about 10 pounds of Lean Body Mass if you are 157 lbs right now. 196.5 is roughly right, math wise, but it may not work that way. I would say shoot for at least 200 lbs before cutting, (and that assumes the 190 end goal) and that would give you a good cushion to hit your goals, plus if you do end up losing some muscle, it won't be that detrimental.
Now seriously at your level, I doubt you will have a problem with all of this. Maintaining 167 lbs of LBM is a lot easier then trying to maintain 210 lbs LBM. And if you are the lean person you say, that fat most likely will melt off of you without a problem. You might find 12% BF too high, and 167 LBM too low for you, but that is for you to decide.