To clarify you are going to ask to swap the battery? It sounds like you are wanting to swap the bike.
From other forums, it sounds like this is a "known" issue for Ducati's and for Li batteries in general. You shouldn't need to trickle charge it (i do hope you got a charge that is rated to charge Li batteries or that is a fire hazard). The easiest way to start your bike when it's cold outside is to just turn on the headlight and let it drain the battery a little then start it (see the video linked below). Li batteries are better in general than lead batteries but you do have to know how to use them.
From other forums, it sounds like this is a "known" issue for Ducati's and for Li batteries in general. You shouldn't need to trickle charge it (i do hope you got a charge that is rated to charge Li batteries or that is a fire hazard). The easiest way to start your bike when it's cold outside is to just turn on the headlight and let it drain the battery a little then start it (see the video linked below). Li batteries are better in general than lead batteries but you do have to know how to use them.