Qatar's industrial capabilities are very limited, but three lots of offsets/participation, etc. will be better than one.
Three different centres of decision making if Qatar does anything to provoke sanctions.
Then why stop at three? Why not order some Sukhois/MiGs and get the Russians onside also?
Three lots of goodwill, invitations for exercise participation and training courses.
You don't need to buy a new fighter type to get goodwill and training opportunities. Countries train with other countries because it is in their strategic interests to do so, and the US/UK/France will train with Qatar so long as it is in their interests to do so. Israel doesn't get invited to Red Flag because it flies Falcons and Eagles.
If goodwill and training opportunities are the objective (and, as you say, Qatar needs to buy another country's kit to get these), then surely buy something that makes operational sense - like a frigate or some tanks. Why buy a third and superfluous fighter type?
And there are some subtle but important capability differences between the three aircraft - Rafale with its heavy precision stand off weapons capability and recce pod. F-15QA with a great payload/range capability. Typhoon may bring with it some Mission Data capabilities for the QEAF, as well as Meteor, Brimstone and Storm Shadow.
Each of these aircraft have essentially the same roles and the same capabilities, albeit with some minor differences as to what weapons types they currently carry. If a country wants a particular weapons capability, then it makes more sense to pay for the OEM to integrate it onto an aircraft they already have or have selected, rather than to go out and buy an entirely new type again.
That said, under current planning the Typhoon is to get the same heavy precision stand off weapons capability and recce pod capability as the Rafale now has, so again why buy them both? And what does Qatar need with the F-15QA's increased range (and if range
is important, they have MRTTs to extend the range of the Rafale or Typhoon).
And the argument that money is no object to Qatar doesn't really hold either, as they could use that money to pay for the weapons integration rather than buy an unnecessary third fighter type.
There is nothing logical about Qatar's decision to field three fighter types.