This is absolutely possible. I did this with Sofi in a roundabout way on one of my last purchases. You do have to be careful in that Sofi will not give you a personal loan to use as a down payment or for the purchase of a property, but they have no problem lending for home improvements, and as you stated, they don't ask a lot of questions. You already noted some of the pitfalls but you'll also want to take into account that your credit will take a hit as the loan is unsecured, and your DTI will look pretty rough because of the payment amount. I'd advise talking to your refinance lender first. Though I'm guessing you could get around most of these problems by signing a letter from your lender that promises to payback the personal loan with the proceeds from the refi.
@Chris Lohmeier If you don't mind me asking, how hard did the loan affect your credit? I'm thinking about doing the same thing with a Sofi loan and with a credit score above 800, I'm worried I'll sink down to 750 or below.
@Alexa K. Between the Sofi loan and the new mortgage on the property I took about a 30 point hit.
Some disclaimers though, the Sofi loan only made up about 5% of my total open balances, and while it did drop my average age of history it didn't bring the average down by more than a year. I'd assume if the loan had more of an affect on these two categories it could lower your score by a bit more.
I'll also mention that this was three months ago and my score has already bounced back by about half of the drop already.
Hope that helps.