SQL Update a Single Row Multiple Times Using 2 Data Sets

Question

I'm working in tsql and have an issue where I need to do multiple updates to a single row based on multiple conditions. 
By Rank_
If the column is NULL I need it to update no matter what the Rank is.
If the Ranks are the same I need it to update in order of T2_ID.
And I need it to use the last updated output.
I've tried using the update statement below but it only does the first update and the rest are ignored. Here is an example of the data sets i'm working w/ and the Desired results. Thanks in advance!
update a
set Middle = case when a.Rank_> b.Rank_ OR a.Middle IS NULL then ISNULL(b.Middle,a.Middle) end,
LName = case when a.Rank_> b.Rank_ OR a.Lname IS NULL then ISNULL(b.LName,a.LName) end,
Rank_ = case when a.Rank_> b.Rank_ then b.Rank_ end
from #temp1 a
inner join #temp2 b on a.fname = b.fname
where b.T2_ID in (select top 100% T2_ID from #temp2 order by T2_ID asc)

Answer 1

The Merge clause actually errors because it attempt to update the same record.  I think this CTE statement is the closest I've come but I'm still working through it as I'm not too familiar w/ them.  It returns multiple rows which I will have to
insert into a temp table to update since the resulting row I need is the last in the table.
;WITH cteRowNumber
AS(
Select DISTINCT
Row_Number() OVER(PARTITION BY a.LName ORDER BY a.LName ASC, a.Rank_ DESC,b.T2ID ASC) AS RowNumber
,a.FName
,a.LName
,b.LName as xLname
,a.MName
,b.MName AS xMName
,a.Rank_
,b.Rank_ AS xRank
,b.T2ID
FROM #temp1 a
inner join #temp2 b
ON a.fname = b.fname
), cteCursor
AS(
Select a.RowNumber,
a.Fname
,a.LName
,a.xLname
,a.MName
,a.xMName
,a.xRank
,a.T2ID
,CASE WHEN a.Rank_ >= a.xRank THEN ISNULL(a.xRank,a.Rank_) else ISNULL(a.Rank_,a.xRank) end AS Alt_Rank_
,CASE WHEN a.Rank_ >= a.xRank THEN ISNULL(a.xMName,a.MName) else ISNULL(a.MName,a.xMName) end AS Alt_MName
,CASE WHEN a.Rank_ >= a.xRank THEN ISNULL(a.xLName,a.lname) else ISNULL(a.LName,a.xlname) end as Alt_Lname
FROM cteRowNumber a
where a.RowNumber = 1
UNION ALL
Select crt.RowNumber
,crt.FName
,crt.LName
,crt.xLname
,crt.MName
,crt.xMName
,crt.xRank
,crt.T2ID
,CASE WHEN Prev.Alt_Rank_ >= crt.xRank THEN ISNULL(crt.xRank,Prev.Alt_Rank_) else ISNULL(Prev.Alt_Rank_,crt.xRank) end AS Alt_Rank
,CASE WHEN Prev.Alt_Rank_ >= crt.xRank THEN ISNULL(crt.xMName,Prev.Alt_MName) else ISNULL(Prev.Alt_MName,crt.xMName) end AS Alt_MName
,CASE WHEN Prev.Alt_Rank_ >= crt.xRank THEN ISNULL(crt.xLName,Prev.Alt_Lname) else ISNULL(Prev.Alt_Lname,crt.xLName) end as Alt_Lname
FROM cteCursor prev
inner join cteRowNumber crt
on prev.fname = crt.fname and prev.RowNumber + 1 = crt.RowNumber
SELECT cte.*
FROM cteCursor cte