Seems to be your current process is working so it'd be a gamble to switch it up.

I have a unit listed for rent in Chicago and I've received several highly suspect or fraudulent applications.

You are gambling when you buy a property that is costing you money every month to own it and your banking on appreciation. 

These strategies must consider the strategic use of loan proceeds and be executed well in advance of there being a need for them, so as to avoid being seen as fraudulent transfers by the courts. 

Matt  my surfer dude buddy.. this is simply post GFC were the mantra is cash flow is king and appreciation is luck .. and well many many people believe that.. or cash flow is the only thing and appreciation is gambling... but if I look at the biggest hits of my career its been on Non cash flow assets IE NO cash flow or negative but multi million dollar paydays. 

They collect these submissions and if something pops up in your area, and your home potential fits their needs, they will typically do a screening, need more information and if you get selected you will begin talking through pricing, contracts, etc.So if you are switching just for the sake of insurance companies, then it would be a bit of gamble becuase these take a lot more time to foster relationships and build trust.

It's a gamble!

I'm willing to take the gamble on them leaving as I will be able to rent them for $400+ more per month with new tenants. 

That is a bit of a gamble.

Keeping that in mind, the majority who overstay their non-immigrant visas will not have social security numbers unless other factors apply or they obtained those cards fraudulently.