I always took offering more than 1-2 months rent as a red flag.

We're there any red flags on their application? 

However, I imagine this would raise some sort of red flag, and the bank manager may or may not do something.

I'm late to the party but your original post raises red flags to me.

After the Facts and Features section, there is a list of drop downs in large, bold font. 

A 3-day cancellation notice, informing you of your right to cancel your contract if you signed the agreement in your home, or at a place other than at the contractor's office or business;All warranties on the owner's rights; A statement warning, in ten point bold type or larger, directly above the space provided for the signature, the following: DO NOT SIGN THIS CONTRACT IF THERE ARE ANY BLANK SPACES.Whether any lien or security interest is on the residence as a consequence of the contract.An enumeration of such other matters upon which the owner and contractor may lawfully agree.Any other provisions otherwise required by the applicable laws of the Commonwealth.A clause informing the owner about: any and all necessary construction-related permitsthat the contractor must obtain such permits.that owners who secure their own building permits or deal with unregistered contractors will not be eligible to access the Guaranty Fund.Acceleration of payment: No contract can contain an acceleration clause that would require any part or all of the balance not yet due to be declared due and payable because the contractor deems himself to be insecure.

If the property has stabilized occupancy, shows a good history of cash flow / management, you don't have any red flags on your PFS, and there's nothing funky going on with the property, you should be able to get 25% down.  

Just got the inspection.Would love some help looking for anything that may be a red flag.The house is a 3x1 single family built in 1970. 

It is a red herring established by the system to protect tenants and legally allow them to stall rent payment. 

My hope is that if a cleanup is done, and it's on record and certified as A-ok by all the appropriate agencies - then it won't be any red-flag for a future buyer.