This is a question for Expatriates or perhaps (if asked in general terms) Law.

Is "not wanting to go back" sufficient grounds to claim asylum?

Your brother will need a much better reason than ‘not wanting to go back’ in order to apply for asylum. Why Italy? He’ll end up in Albania

The invitation form contains a warning about possible fine/prosecution/imprisonment if anything in it is incorrect or a knowingly false statement.

This seems like something you need to discuss with an immigration lawyer, not just a bunch of amateur travel enthusiasts.

@WeatherVane Obviously not but the question doesn't imply that it is, how is that relevant to the OP's predicament?

Why do you think your husband admitted that he is responsible that your brother will go back and what would that entail? Some nordic countries try to make sponsors financially liable for claims to public support from their relatives but beyond that it's not clear how you could force someone to leave or be responsible for their actions in a modern legal system. In any case, that's not what the Schengen invitation form is about and there is no such implication in the documents shared by @Traveller

IMHO the impact on you (apart ev. financially responsibilities as written by Relaxed) would be that your will be less trusted on further invitations. -- Other stack exchanges may be relevant on how to handle "personal issues" (convincing your brother to return as he promised).

While we don't know your brother's circumstances, and don't know whether asylum is in any way justified or if it's just a way to try to game the system, but he should note that asylum requests cannot be made in any country they want. Under the Dublin Regulation, they have to apply in Sweden since Sweden issued a visa for him, and if they apply elsewhere they are likely to be sent back to Sweden. Rules may be different if he is a minor or has immediate family (spouse, children) in another country. Though I wonder why he would prefer Italy over Sweden...

@jcaron I am not sure how much is going to change with the new AMMR but that's not at all how the Dublin system works. In fact, Dublin requests only come into play after a person has lodged an application. The system's entire point is to ask another country to take care of a refugee, precisely because people actually can apply anywhere they want.

Importantly, Dublin regulations explicitly allow any state to process any application they see fit (article 17 of the Dublin III regulation). There is no obligation for either the refugee or the state where the person is seeking to have their status recognized to do anything based on the Dublin regulation. Because states try very hard to get rid of refugees, the obligation is on the responsible member state, not the state of application.

I think (but IANAL - it feels a little weird to say that in a comment aimed at regulars) @Relaxed is right about how the Dublin system works, but in this case the end result is that Italy would be allowed to (and probably will) refer him to Sweden (if he had a layover on the way to Sweden that might come into play and change that).

Does your brother actually have a reason for asylum? Meaning, is he being persecuted? If not then an application for asylum will cause massive trouble for both him and you.

@DJClayworth What specific negative consequences do you suppose an application could cause?

Thrown out of the EU, banned from future applications, inability of the OP to guarantee other visas are all possible.

@DJClayworth Merely for lodging an application that is ultimately deemed frivolous? Removal is precisely what an asylum application is trying to fend off and once your application is thrown out you're back to square one, at the beginning of the removal procedure. Something like a 5-year ban seems actually more likely if you are found staying illegally without having lodged an application, which is the alternative people with little perspective at securing a long-stay visa actually face.

@Relaxed: It all depends on the specifics. The application will probably require the applicant to make (or agree to) specific statements about their supposed fear of persecution in their home country. If those statements turn out to be lies, then most immigration regimes will have absolutely no problem holding that against the applicant as if they had lied on a visa application.

@Traveller (So far) only refugees picked up at sea may end up in Albania.

If you and your husband have not made any untruthful declarations, then you shouldn't have to face any consequences.

What is the citizenship or residency status of your husband? I suppose he has solely Pakistani citizenship — is this accurate?