I'm running a campaign set in the 1350s DR and the party has traveled to Zhentil Keep. The party has legit papers from High Hall courtesy of a Zhent merchant they accompanied to the city. They are not aligned with either major faction of the period, the Zhentarim or Harpers. Ruins of Zhentil Keep lists many inns in different parts of the city, in the chapter Zhentil Keep - Then. The chapter Zhentil Keep - Now of Ruins of Zhentil Keep is not applicable to this question as the destruction of most of the city happens in 1368 DR.
The party is mostly good and mostly human with one half-orc (Mistress Mara Trollsblood, AL LE, a minor encounter from Doom of Daggerdale upgraded to NPC), and an elf wizard who is disguised as a half-elf bard. I'm using Basic D&D with old 1E and 2E Forgotten Realms books including the old Grey Book (FR0) and Ruins of Zhentil Keep. Probably I'm going to switch to 5E once I finish reading the books, but keep the 1350s DR time period. Nothing in this question is edition specific which is why I marked it as lore.
Would they be restricted to the Foreign Quarter or could they lodge elsewhere in the city?
What restrictions could there be on their lodgings as neither Zhent or Zhentarim?