I don’t know every shliach – they each have to deal with each situation according to halacha and with tact- and they are trusted to do so on their own – there is no shliach manual that goes through every conceivable situation –
That being said, I know shluchim who for that reason don’t eat at anyone’s houses as a policy so they don’t get into sticky situations. Others will eat by mekuravim / completely frum ones, and in the case above, some will accept but make it very clear whats acceptable such as eating in Plastics, foods that would work like bagels and lox with acceptably hashgacha that are opened only after the Rabbi arrives
We invited an intermarried couple for Shabbos and they wanted to invite us back, but we explained the kashrus issues would be too complicated for a shabbos meal. They still wanted to host back so they ended up paying for a catered shabbos from a caterer I chose and they brought wine and pastries from a local acceptable bakery…