Although option 3 doesn’t have any formal error that would be listed in a textbook, it doesn’t sound natural to my (American English) ear.
First, the progressive mood sounds very unusual here. I would more likely say something like, “I wish I had more kindness for him,” “I wish I could be more charitable,” or following the prompt more closely, “I wish I were kinder to him.” Using I were instead of I was or I had been is enough to mark the verb as present subjunctive.
Also, most native speakers would say, “wished she was” unless we were being very formal. For instance, a Google search of the White House website shows roughly 960 hits for “if he was” versus 129 for “if he were,” and the “if he were” hits were the most serious statements (such as the President saying “if he were here” of his dead son in a Memorial Day address, or threatening consequences against a foreign leader “if he were to invade”).
For whatever reason, “If I were being kind to you,” sounds more natural (but sarcastic) to me than “I wish I were being ....”
“Were being” followed by a past participle is almost always a past passive, such as, “steps were being taken.” I’d speculate that might be how we came to avoid that construction here. In this case, with a singular subject and an adjective that cannot double as a participle, the sentence is unambiguous, but with a different subject and adjective, “were being” would be confusing. For example, “I thought they were being frustrated,” would be ambiguous: it could either be simple past passive (I thought that something was frustrating them in the past) or present progressive subhunctive (I thought counterfactually that they were feeling frustrated now).