After quite a bit of fiddling around, I found out I can make use of CONCAT, as long as I cast it properly:

SELECT * FROM my_table
WHERE my_col->'my_field' @> ('["' || :dyn_val! || '"]')::jsonb;

This behaves as expected: it uses my dynamic string, encloses it in with the right syntax and once casted as jsonb, can be used with the @> operator.