Bug: "Invalid column name" after sp_recompile for valid column in custom data type

Has anyone ever seen anything like this?

Yes. I've seen this sort of thing when there are #temp tables named the same in both parent and child scopes.

See this example DB Fiddle https://dbfiddle.uk/H05-MO65 

Line numbers often just point to the statement that has the problem rather than the actual specific line so per the additional code you showed in the https://stackoverflow.com/q/79769659/73226 this statement is, in fact, a couple of CTEs followed by 

INSERT INTO #Islands (KEY_1, KEY_2, KEY_3, START_DATE, END_DATE)

So you could definitely see this type of thing if the outer proc also has a temp table called #Islands but without the KEY_3 column