That is accepted, that's why you can't report it.
You can theoretically report that the translation is unnatural, but it is better to write here, actually.
I think both are possible. Do you have a reason to believe the answer above is wrong? I think it is grammatically correct: "When we talk about present regrets, both wish and if only are followed by the past simple tense. The past tense emphasises that we are talking about something ‘unreal’" https://learnenglish.britishcouncil.org/en/intermediate-grammar/wish-and-if-only
I think I didn't have the option to use would because I was composing the English from words provided. Or I was going from English to Czech, I can't remember now. This is quite a fine difference but for me the sentence given in English with past simple does not sound natural in this case, though of course context is everything and it might be possible to find one that would make it reasonable. According to 'Practical English Usage' M Swan, OUP 1995: wish + past simple 'expresses regret that things are not different and refers to situation that are unreal, impossible or unlikely.' The problem is that 'staying' is not really a situation. So 'I wish he was there/somewhere else.' would be OK. Whereas wish + would is used when you want someone/thing to do or not do something, or to be willing to do something. "I wish he stayed there" would sound like 'he' maybe habitually visits 'here' and the speaker doesn't like it. But then that doesn't go with 'forever'. 'I wish he would stay there forever' is like saying 'I want him to stay there forever'. Which seems the most likely sense.