Excerpt of text from Timatic below. Looks like transit is only allowed at MXP Malpensa T1 which is where you are self-transferring. If you are considered in Transit, you should not need a Transit Visa since you've a USA issued F2 visa. I presume it has not expired but anyways it has been used so you should be good there.

Your deal-breaking issue next week will be whether you will in fact be considered in "Transit" in Italy because that is the only way you will not require a Schengen visa. I understand you have booked a "Self-transfer" option through BudgetAir. What exactly does this mean? Will United in St. Louis issue you a boarding pass through to Pakistan? Or will they only give you boarding passes to MXP but still be able to see the Kuwait Airways flights in their system? Because you say there are two separate tickets, I am not sure what the BudgetAir itinerary shows: whether your e-ticket is from St. Louis to Islamabad — which is the only way you would be considered in Transit — or St. Louis to MXP and MXP to Islamabad in which case UA will likely only see their flights to MXP. Your problem will be if UA in St. Louis cannot see the onward flights out of MXP then they will rightfully consider your journey as terminating in MXP and their system will not issue you a boarding pass unless they enter a valid authorization to enter MXP and you will not even get to Newark. Having an onward ticket out of Italy may not be sufficient.

So bottom line is that this is a hugely risky gamble you have taken and anyone, anywhere in the world should avoid a self-transfer on separate tickets if the location in which they are planning to perform the self-transfer requires a visa for their passport type. It is the passenger's responsibility to check visa requirements prior to purchasing tickets. I think trying to get a Schengen visa at such short notice from your closest Italian consulate in Chicago is also tricky. You should immediately call United Airlines, Kuwait Airways and Budget Air and go from there to check whether United can see the Kuwait flights in the itinerary and will consider you a through passenger all the way to Islamabad.

Here's your other issue: I don't know if there is a transfer desk in the International Transit area of MXP. Yes I understand you are flying without checked bags — but these self-transfer itineraries will probably assume you have checked bags. Which means BudgetAir is expecting you to enter Italy, pick up your bags and recheck with Kuwait. There may not be an interline agreement between UA and KU. These questions can only be answered by United Airlines, Kuwait Airways and Budget Air so please call them, and avoid booking any self-transfers again in locations for which you will ordinarily require a visa to enter.

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Italy TRANSIT-
Type: Critical
Visa Requirements
-Passengers must have a valid visa issued by Italy.

Exemptions for 3rd Country Visa

Passengers can enter with a “C” or “D” visa issued by a Schengen Member State for a maximum of 90 days in a period of 180 days.

TWOV for 3rd Country Visas

Nationals of Pakistan can transit through Milan (MXP), Rome (FCO) or Venice (VCE) for a maximum of 48 hours with a visa issued by Aruba, Bonaire, St. Eustatius and Saba, Bulgaria, Canada, Curacao, Cyprus, Ireland (Rep.), Japan, Romania, St. Maarten or USA. An expired visa is accepted if:
- it has already been used and was valid on the day of departure; and
- the passenger is returning from the country that issued the visa.

Type: Ok
Transit Information
-Passengers can only transit within Terminal 1 at Milan Malpensa Airport (MXP).

To confirm if a visa issued by the USA has been used, it can be verified at https://i94.cbp.dhs.gov/home . Click on the View Travel History tab on the top of the page.