is it suitable to simply store the uid's in a text file separated by a "/c" or space or other token so that the second Arduino can search for the uid?
Yes. It makes more sense to me to store each UID on a new record/line, so that all records are fixed length, which makes searching faster.
If this is done say 1000 times with the data appended to the new uid
With what data appended to the UID?
Using fixed length record, each consisting of 2 characters ("in use" and a space) followed by the UID and a CR/LF, you could quickly find the appropriate UID record. Then, set or clear the in use flag.
That way, the number of records will never exceed the number of tags, and there will be no empty records.