As long as you‘re resident in Sweden and it‘s your funds, you have to pay them anyway.
If it’s your money that you‘d be trading with, only in the name of your girlfriend - that proposal basically amounts to illegal tax evasion. And it would very likely violate IBKRs agreement (though you could probably have a joint account with your girlfriend or even administer her trading on her money).
Regardless of actual reporting, if you move from Sweden to Switzerland, you will (generally) cease to be a Swedish tax resident and not pay taxes in Sweden anymore.
(there may be some transitional rules for the year you’re moving, some countries do have exit taxes or particular, limited exceptions for emigrants, that are basically trying to counter emigration that is only intended avoid taxes, but these are rare cases and unlikely to apply for personal investors that genuinely move to another country for employment or work. Keep in mind that the EU intends to guarantee your right of free movement in Europe and remove impediments to that).
I don‘t know how IBKR’s EU entities deal with that once you‘d leave the EU. In any case, unless you‘re classified as a professional investor, you they won‘t let EU residents trade them.
Taxation is (generally) based on your current tax residence. NOT on your country of residence at the time of opening up an investment account and NOT on the country your account/bank or broker is located.