I'm wondering if I should let him live without the lease as he's increasing the value of the property while still paying rent.

I had several properties I bought in 2008 that subsequently lost 1/2 their value.   

Obviously, the home could be sold as the old sq ft and the investor/buyer can just go and permit it for the increased sq ft,  but I would think that the county automatically will go and assess the home as soon as it changes hands since this the county has probably seen this before. 

That way your monthly mortgage (and the interest you are paying) doesn't increase but you access to the equity in your home (some banks are giving up to 95% LTV).

If you want to increase cash flow then we can talk about reducing debt.

The properties were bought with 100% leverage, so once the loan for the down payments are paid off in a few years, the cash flow will increase pretty dramatically. 

(There is uncertainty though how much Feds will increase interest rates next 6 months.) 

You could go to your insurance company and request they lower the replacement cost or increase your deductible... if they don't want to write a policy under those conditions, you can look for another company.

If I aim at 12%, even a rent increase would not offset the expenses enough to hit 12%.

I give a 30 day notice for rental increases.