Scandinavian Airlines and Lufthansa have round-trip flights from Los Angeles (LAX) to Frankfurt, Budapest, or Gdansk (FRA, BUD, GDN) for $425-$534 with a connection. Flights October to March. Gdansk is the cheapest, Budapest is the most expensive.
Carry on bag with a maximum weight of 18 lbs is allowed for free on SAS. You can avoid seat selection fees by accepting the randomly assigned seat at the gate. Upgrading to a higher fare class that includes a full-size carry-on bag for $28 may be cheaper than booking the discount fare with the added carry-on bag fee (for about $60).
Lufthansa “Light Fares” allow for a carry-on bag (17.6-lbs @ 21.6 x 15.7 x 9 inches) and also a very small personal item such as a laptop bag (11.8 x 15.7 x 3.9 inches). The weight limit will probably be the limiting factor for those of you who wish to avoid bag fees. Prioritizing heavier items in your “personal item” may be a good way to lighten your carry-on bag. You can avoid seat selection fees by accepting the randomly assigned seat at the gate.