tsearch strategy for incremental search

I am trying to implement an incremental search engine. The service
should start searching when the user has typed at least 3 characters.
I am thinking of using the following strategy:

a) Create a function string_to_three_char_tsvector(str text) that
would generate the tsvector composed of the three-letter lexemes that
begin all the admissible words within the parameter str.
b) Using this function, create an indexed tsvector column: three_char_index.
c) Given the query string query_string (assume query_string containing
at least 3 characters):
SELECT *
FROM mytable, plainto_tsquery((string_to_three_char_tsvector(query_string))::text)
AS query
WHERE three_char_index @@ query
AND text_field LIKE '%' || str || '%';

Once I've narrowed the field of possibilities down to the correct
3-letter lexemes, there are fewer than 100 lines to search through
with LIKE. I could even repeat the exercise with 4-letter lexemes if
these numbers were to grow or if I needed the extra boost in
performance.

So, two questions to postgres/tsearch experts:

1) Does that seem like a decent overall strategy?

2) About the function string_to_three_char_tsvector(text), I cannot
think of an elegant way of writing this. Is it possible to do better
than the following:

str => cast to tsvector => cast to text => for each lexeme-string,
take first-three-char substring => concat back together => cast to
tsvector

Is there a nice way of performing the middle operation? Like
splitting the string to an array...