mileshound welcome to the QR forum.

Because your AA reward flights are on AA ticket stock you will not qualify for the free 96 hour tourist visa. When the scheme started it was permitted but alas no longer.

Your plan is a good one so I say go for it.
Do remember that because your transit is under 24 hours you cannot check your baggage to DOH only. It will be through checked to your final destination and you will not be able to access it in DOH. Everything you require during your transit needs to be carried in your hand luggage.