SQL Function for updating column with values

You want to proportion the "temp" to get equal amounts of "A" and "B".

So, the idea is to count up everything in A, B, and Temp and divide by 2. That is the final group size. Then you can use arithmetic to allocate the rows in Temp to the two groups:

select t.*,
       (case when seqnum + a_cnt <= final_group_size then 'A' else 'B' end) as allocated_group
from (select t.*, row_number() over (order by newid()) as seqnum
      from t
      where group = 'Temp'
     ) t cross join
     (select (cnt_a + cnt_b + cnt_temp) / 2 as final_group_size,
             g.*
      from (select sum(case when group = 'A' then 1 else 0 end) as cnt_a,
                   sum(case when group = 'B' then 1 else 0 end) as cnt_b,
                   sum(case when group = 'Temp' then 1 else 0 end) as cnt_temp
            from t
           ) g
     ) g

SQL Server makes it easy to put this into an update:

with toupdate as (
      select t.*,
             (case when seqnum + a_cnt <= final_group_size then 'A' else 'B' end) as allocated_group
      from (select t.*, row_number() over (order by newid()) as seqnum
            from t
            where group = 'Temp'
           ) t cross join
           (select (cnt_a + cnt_b + cnt_temp) / 2 as final_group_size,
                   g.*
            from (select sum(case when group = 'A' then 1 else 0 end) as cnt_a,
                         sum(case when group = 'B' then 1 else 0 end) as cnt_b,
                         sum(case when group = 'Temp' then 1 else 0 end) as cnt_temp
                  from t
                 ) g
           ) g
      )
update toupdate
    set group = allocated_group;