Which affiliation should I use? I am a postdoc working in one university till 1. July. I am leaving for another university after that. During my stay at my current work, I wrote a paper as a first author for a conference that will kick off in September.

Which affiliation should I use? Taking into account that the conference travel will be covered by the new employer, however, the paper is written at my current university.

By reading this article, it looks like using the two affiliations is one approach. But I'm not quite sure if the new employer will be happy with this.
You are conflating funding sources with affiliations when they should be discussed separately.

The affiliations tell you where the work was performed. The work was performed at the old institution, so it must be listed as an affiliation, and should be the primary affiliation. If you have done work on the presentation at the new institution—for instance, used your research time to prepare the presentation and rehearse it in front of colleagues—then you can list the new institute as an additional affiliation. It should be listed as a "current address" in any event.

As for the funding issue, since they're providing the resources for you to travel, it can be acknowledged when listing anybody else who sponsored the research alongside those other sources.