Heuristic for weighted maximum independent set in graph with ~$2 \times 10^5$ nodes and $|E| \propto |V|$
Unfortunately, (weighted) maximum independent set is very hard to approximate. You might be able to do a bit better if you can analyze the graphs in your application (perhaps they are not truly arbitrary).

In any case, luckily your graphs are quite small (200 thousand vertices or so). A naive algorithm (say a greedy one) will run quite fast, but might be far from the optimum. Especially since several hours (or even days) of runtime is fine, I'd experiment with a genetic algorithm. 

Being a rather central problem, there are studies into this as well. For a starter, one could look at [1]. In practice, I would expect one to be quite happy with such an approach.



[1] Hifi, Mhand. "A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems." Journal of the Operational Research Society 48.6 (1997): 612-622.