How to deal with results that are given as exercises
In mathematics, if you cite a result, you should really be able provide the proof if somebody asks for it. So I would strongly recommend against option 3.

One thing you might consider is writing the author of the textbook, explaining your situation, and asking if he's willing to send you a proof. Then you can cite it as 


  problem 8.35 in textbook, proof provided by private communication with [author]


even if he would rather not have a proof of the exercise published. 

Of course, if he replies "I think I had a proof when I wrote the textbook, but I've forgotten it now," you might need to come up with a proof in some other way.

There is at least one exercise in a textbook whose proof is highly non-trivial, and was an open problem for several years before a journal paper giving the proof was published (and the only indication in the textbook of this is that part (b) of the problem is "optional").