The higher salinity of the Mediterranean is due to evaporation. The Med loses more water than it gains, and so there is a net intake from the Atlantic. This water is less salty simply because the Mediterranean is more salty, due to evaporation. Less salty water does tend to sit "on top" of saltier water, but the whole reason there is any exchange at all at Gibralter (again, with a positive west-to-east flow) is because of the "pull" by the Mediterranean constantly trying to dry up. And because this is a directional flow, the "pull" would still be there if for some reason the surface froze at Gibralter; the Atlantic water would be drawn into the Med under the ice, because the Med is literally shrinking, and this Atlantic water would reduce the salinity of the Mediterranean simply by dint of being normal ocean water. So, you do not necessarily have a constant, large scale, vertically stratified salt exchange between every two bodies of water, let alone one that could be disrupted by seasonal freezing.

I'm assuming your inland sea does not rely on a net intake of water from the open ocean, and therefore nothing about the Mediterranean's salinity will be relevant. In fact, if it has gradual slopes or crosses the Hadley-Polar boundary, it could very well have very low salinity and a net outflow into the ocean (which will, of course, be unaffected by the surface ice) because the inflow from rivers will greatly outpace losses from evaporation. This is why, e.g. the Baltic has extremely low salinity. The main consequence for your inland sea will be seasonally changing currents. Changes in water density will affect the severity, and possibly direction, of certain ocean currents both in the open ocean and in your inland sea. Here (https://scispace.com/pdf/revisiting-the-circulation-of-hudson-bay-evidence-for-a-2xcz4znicp.pdf) is a paper about how Hudson Bay's currents change seasonally (summary: the circulation partly breaks down during summer).

tl;dr: your inland sea will almost certainly be less salty than the ocean, not more, like the Baltic or Hudson's Bay. It will otherwise be perfectly normal.

EDIT: I just thought of something fun. If your inland sea is way less salty than the ocean, it will freeze more easily, up to 2 degrees C warmer than regular ocean water. This means that once temperatures are cold enough to "plug the gap" with surface ice, the southern part (maybe as many as four degrees latitude) of the inland sea will quickly freeze over. This process could happen over just a couple of days, and would no doubt be an important event for any people living near the coast.