Descending the Grass River from Paint Lake would take you to the Nelson, just below the Kelsey generating station and just upstream of Split Lake.
If you look back through this thread -
http://www.myccr.com/phpbbforum/viewtop ... 13&t=40590you will see a couple of posts about a route from near Split Lake on the Nelson to Churchill via Waskaiowaka Lake and the Little Churchill and Churchill Rivers. Whether the person inquiring about a route ever made the trip, I am not sure.
In the Manitoba trip reports forum is a report on the Little Churchill / Churchill section by "Rappi".
Without a great detour to the west, I don't see another option for reaching the town of Churchill from your position via the Grass / Nelson. That said, this would be a very serious undertaking and I think the crossing to Waskaiowaka could be problematic in a sea kayak.
-jmc