First, it is unlikely that any scope will change its settings. It is very likely that the gun will change its point of impact over time in storage. If you dismantle the gun to clean and put it back together, it will change its POI. Wood will absorb moisture, or dry up and warp just a bit. But the scope will likely not change its settings, no matter which one you buy.
Second, for those reasons, don't go hunting without checking the sights on your gun, no matter what scope is on it.
Third, for $200 there are lots of scopes that will work, at least for a while. What you buy with more expensive scopes is good optics, reliable adjustment, and long life. I have had very good luck with Leupold scopes, but getting one (at least here in Canada) for less than $200 is hard. They are not the only scopes that work, but I recommend them to anyone who asks.
Lastly, the sight on a gun is a vital part of how much you will enjoy the gun, and is also vital for its function (shooting accurately). A good one will last a lifetime, so spend enough to get a good quality scope, and it will seem very cheap indeed in about 15 years.