Depending on the quantity of value you think you can create through the renovations, I wouldn't worry about the 20% down unless your renovations wouldn't increase the value of the home enough to cross that 20% threshold once complete.
I don't know about all loans, but my loan(non FHA) allows me to hire an appraiser after 12 months to re-eval the value of the home. If the home is worth more, then my loan will recognize that new appraised value and adjust the % of ownership accordingly.
Ex: I practically stole my house, but put down less than 20%. I plan to get an appraisal on it next fall as it was appraised at approx 25K more than what I bought it for and the houses for sale in the neighborhood that are comparable are asking $10-$15k above that. So, if the neighbors aren't crazy and the last guy who did the appraisal isn't Fake Sugar Dick (WARNING, NOT THE REAL SUGAR DICK!), I could spend $450 next fall for the appraisal, file it with the bank, and all of a sudden my % of equity will be enough to cross the 20% mark and my PMI will drop off.
All that said, PMI isn't very expensive. Mine is like $80/mo.