Why is relative sea-level falling in Hudson Bay?
The area is experiencing post-glacial isostatic rebound.

Much of Canada was covered in an extensive ice sheet in the last glacial period (the 'Ice Age'), from about 110 ka until 12 ka. The ice in the Hudson Bay area was among the last to melt:



A thick ice sheet depresses the crust (the lithosphere), making a small dent in the uppermost mantle (the asthenosphere) in the process. Well, not that small: p 375 in Gornitz (2009, Encyclopedia of Paleoclimatology and Ancient Environments) says it could be 800 m for a 3000 metre-thick ice sheet!

Since the asthenosphere is highly viscous, it takes a long time time for the depression to 'bounce' back up. This map from Natural Resources Canada shows the current rate:



Since global sea-level is currently rising at about 3 mm/a, a local uplift at this rate will break even. Anything more will result in relative sea-level fall, as we see in Hudson Bay (as well as in Scandinavia, the UK, Alaska, and elsewhere — this map is wonderful).

Interesting, for geologists anyway, is the sedimentological record this leaves. I love this example of raised beaches and a small delta experiencing forced regression on the shores of Hudson Bay:



Last thing — you asked:


  Would it make sense for it to recede at the level that it is
  receding with sources of freshwater relatively close?


Since Hudson Bay is connected to the world's ocean, mainly through Hudson Strait, the runoff into the Bay has no measurable effect on the water level.

Credit
Ice retreat map by TKostolany, licensed CC-BY-SA. Rebound map by NRCan, free of copyright. Google Maps image contains own credit.