I encountered a related issue, and @amapanda_ᚐᚋᚐᚅᚇᚐ shared this topic with me. Like a good number of people have suggested, I think updating the tagging is the solution. However, I don’t think waterway is the correct tag, since there’s literally not a river in a lake: these rivers-in-lakes exist only to facilitate hydrologic modeling/analysis. Perhaps modeling:waterway=river would work best.
The tag change would let renderers know the tagged geometry shouldn’t be displayed at all, preserving the “on the ground” rule, allowing the use of the name tag (avoiding the river name appearing in the middle of the lake), while preserving the network connectivity, without having to treat the lake perimeter as the connecting geometry.