Both "I asked him to come back" and "I asked him to return" sound natural enough in British English, although I understand the subjunctive "I asked that he return" doesn't sound as stilted in American English.
I think that it sounds a bit stilted and very formal in American English as well.
That is British English in which the verb is inflected or substituted by "should + base verb".
Yes, British English. AmE has maintained the classic subjunctive form in which the base verb is used in the present and past.
DL's sentence is a demand, not a question.
• I asked that he come back/return. (no "s" on 3p singular)
• I asked him to come back/return. (infinitive form)
Subjunctive construction in English: no "s" on 3rd person singular (base verb).
I asked that he come back.
Pedi que ele voltasse.
"I asked that he would come back" or "I asked that he'd come back" also seems overly formal, but I thought they sounded more subjunctive and they weren't accepted. I'm not sure, even with years of language study, I understand anything about the subjunctive. Despite many readings, lectures and of course Duolingo lessons.