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I just found very nice solution!
1. Lets assume for a moment, that we load additional integer column (i named it "priority"). It has values corresponding to header 4, but without letter a. So, it has values 1 for a1, 2 for a2, etc.
2. Lets assume, that we are able during reload create expression body like this (and for exaple put it into variable value).:
pick(min(priority), max({<priority={1}>}[Header 5]),max({<priority={2}>}[Header 5]),max({<priority={3}>}[Header 5]),....max({<priority={n}>}[Header 5]) )
(I have no time to prepare load script which will build this expression, but it is not difficult)
in my case expression is:
pick(min(priority), max({<priority={1}>}[Header 5]),max({<priority={2}>}[Header 5]),max({<priority={3}>}[Header 5])),
because in my tests max priority in all popullation has value 3.
3. If we have above done, solution is easy and, should has normal performance.
Take a look on last column in chart.
Have fun with QV 
regards
Darek
Please take expression from comm_header1_5_v3.qvw I attached in one of my responses.
1. column with Header4 is Calculated Dimension:
=aggr(minstring([Header 4]),[Header 1],[Header 2],[Header 3])
So, i simple gives you minimal valye of H4 for combination of H1,H2,H3
2. Header 5 column:
pick(min(priority), max({<priority={1}>}[Header 5]),max({<priority={2}>}[Header 5]),max({<priority={3}>}[Header 5]))
a)
pick(n,ex1,ex2,ex3,...,exm)
n-integer value, in our case n=min(priority)
pick() gives you n'th expression. So, if our priority = 2, there will be ex2
our expression are very similiar to each other:
max({<priority={1}>}[Header 5]) is taken when priority =1
max({<priority={2}>}[Header 5]) is taken when priority =2,
.......
Belive me, even if today you have only 2 priorities, there may be someday 3-rd priority So, be ready for it
If you need some more information, please let me know.
regards
Darek