Splitting conference expenses between authors
Creativity and asking the financially responsible person at both author A's and B's institutions about what is possible (maybe suggesting a few ideas) is key here. In fact there are a lot of options, and if the funding budgets at the two institutions have different regulations it may add flexibility for you to exploit. Some ideas are:


Author A travels to the conference, but in a (partial) mission for author B's institution, so the latter agrees to cover travel cost (lodging and flights). They might be able to buy tickets / reserve a hotel directly, or pay a reimbursement.
Author B's institution handles (and pays) the registration of Author A to the conference - with A's name of course! Or A gets a reimbursement for the registration fee from B's institution.
Author A gets a honorary into his own pocket from author B's institution for preparing and giving the presentation of author B's contribution to the paper, and pays part of the travel cost out of this own pocket (be careful with taxes etc., but if the honorary is not more than the personally covered cost I would hope it's not a problem in author A's legislation).
Author A's institution initially covers all cost, but since it's also partially done for Author B's institution, they issue a bill with the previously agreed contribution.


Universities as well as research institutes should generally have a process for reimbursing someone not employed at the place - it's needed for all sorts of things from academic guest speakers to people participating as test subjects in research projects.