Yeah, the other way is more general: create, then join to a temporary table. (that floor() was just a first try, I should have removed it)

insert into $t values (394, 2, '*Inserted 1st*', :insert_location);

-- update $t set f_sort = floor(f_sort)+1 where f_sort >= :insert_loca
+tion;  -- meh

-- instead:

-- create temporary table
create temp table _t as 
   select f_node_id, f_sort, row_number() over () as new_number
   from (
      select f_node_id, f_sort
      from $t
      order by f_sort
   ) as f
   order by new_number ;

-- update table
update $t t  set f_sort = _t.new_number
from _t 
where _t.f_node_id = t.f_node_id --> join expression
  and _t.new_number <> t.f_sort  --> avoid unnecessary writes
;

drop table _t; --> remove temp table
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It's verbose-ugly but it works without repeating the values.