What services (possibly travel agencies) are out there to help me
figure out where and when I should go on vacation, given some fairly
complicated constraints?

The main factor that I need help figuring out is how to make use of
the large number of frequent flier miles I have on both United and
American Airlines (about 150,000 on each).  I have already contacted
http://www.awardplanner.com/ about this, and they "declined" to take
my request because it was too complicated.

What's complicated about my request is that it appears to be really
hard to figure out how to use these miles to go where we want to go
(Bali).  And it may require deciding to go somewhere else, but we'll
need advice about where is best to go.  (And where our miles can get
us.)  Also, we're hoping to fly business class -- either getting award
tickets or purchasing coach tickets and using miles to upgrade.  But
the restrictions are so complicated and there are so many dates
unavailable, it seems to be next to impossible.

Here's what I'm hoping you can help me find:   a travel agent (or
maybe some other kind of online service) that will spend some time
trying to find the best solution for me.  Someone who will take the
time to understand what kind of place we want to go (exotic beach
location that is beautiful and tranquil), what dates we are available,
when different destinations are in season, what miles we have
available, etc, and then make a suggestion: e.g. "You can go to Phuket
in November instead of Bali in October".

The problem is that travel agents usually make their money from the
commission, and there isn't any for award travel.  Also, most travel
agents I've dealt with really just want to book your travel and move
on; they don't want to take the time to figure out the optimal
solution.   I'd be happy to pay someone an hourly rate to work on this
for me, but I don't know where to begin looking for someone to do it. 
Can you help me find someone?  Maybe it's just a really great travel
agent, or maybe it's a service like AwardPlanner.com (but willing to
do more).

I live in LA but would be happy to use a service elsewhere if I can do
it by phone (or email).

Thanks!!

How complex could it be to book flights? You might be better off
calling the airlines directly and asking them what flights are
available to the destinations you want to travel. That way, the person
arranging the trip (you) knows all of your preferences and
restrictions.  You will probably spend more time trying to find
someone to do this for you and explaining all of your restrictions,
than you would doing the research and booking yourself.