Stephen, this gave me the right path towards finding a solution in my case, the part about "updates values in both tables". Thank you.
I tried really getting to the bottom of why this was happening with some tables and not others, so I duplicated the table I wanted to lookup records in and removed all fields except for the primary field.
Let's call the "parent" table, the Table whose record I want to link, Table 1, and the Table I want to link it to (the Table I'm in when trying to add a record and link the Table 1 record to it), Table 2. Let's call my duplicated Table 1, Table 1 Copy.
Linking a record from Table 1 Copy (which has only the primary field) worked fine.
Then I added lookup fields to Table 1 Copy for the first 6 or so Table 1 fields (the ones that appear when selecting a record on the iPad) to Table 1 Copy - ok, error.
Then I removed an Attachments Lookup field, which was finding photos I would add to the records in Table 2 which would appear in Table 1 through a lookup field. Works fine.
So, the solution was just a question of re-ordering the columns that appear first, and thus which fields appear in the little pop-up box that shows you those fields when you go to link a record from iPad.
For those who don't know, Airtable will show you the Linked Record columns in the order they are set in the first (top-most) view of the Linked Table, unless you choose to allow a record selection from a specific View, in which case that View's column order is followed, and those values are shown in the selection box for the linked record.
If this works for someone else too and they would like to mark this issue solved, that'd be great.
