If your flights are booked as one ticket through United (assuming SEA-ORD-WAW-BUD) then United's baggage rules apply to your itinerary. If you book business class you can check 3 bags up to 70lbs for free (assuming *G status) and each bag after that is $200 each.

This is what I did when I moved to Singapore last year. I checked 5 bags and paid $400.

Flying on LOT or Lufthansa doesn't matter unless you decide to take the SEA-FRA LH non-stop. Then LH baggage rules will apply which is 2 free bags at 70lbs in business. Each additional bag (not guaranteed to even make the flight which I find odd considering the traveler must travel with their bag on international flights to/from the USA) are: $200 per bag.

So let's say you have five bags to check:
UA = $400 excess baggage (if *G); $600 excess baggage if not *G.
LH = $600 excess baggage

Besides the miles being equal to book these flights, I would pick UA as your initial carrier through ORD.

-RM