Increasing the property value would cost us more in taxes than paying the closing costs for a cash-out re-if on the other property. 

Once upon a time appraisers made a decent income relative to living expenses and would do this sort of thorough research, but appraisal fees haven't been tracking with increased costs of living for some time, so it's currently a volume game for appraisers, not really a quality game.

I have a 15y fixed 4.25% already locked in (with the option to increase the loan amount up to about 100k).

I always tell my clients: instead of being frustrated by the rules or certain limitations, you should understand the rules and embrace them for your benefits.

Will the property change even if we do not increase the living sq ft to the building?

Are they not willing to bump up the estimates due to the increased rehab costs?

With rates moving higher, it is likely that your rate will increase in the future.

in which case, massive increases (besides property taxes being high in chicago to begin with)

How can you increase that cash flow?

Here are a few options:1)HELOC for repairs and flip it for profit2)HELOC for more rentals and hold current to pay off HELOC3)Seller finance to reduce tax obligation and receive. a large enough initial downpayment from buyer to use FHA for a primary4) Hold and use cashflow with the specific intent to invest all residual income5) Sell and take "Capital Gains" hit but purchase another property to help offset it6) Transfer deed to an LLC to remove it from Grandma's name (this opens another door of possibilities but may trigger tax increase due to transfer of ownership)I can go on, and others with different approaches can add to this list as well.