I'm a Mortgage Consultant by day and I have to say you have not given all the facts in order to answer your question.
What rate you can get is affected by a host of things like your credit score, exactly how much equity you have, escrow (no escrow), and the loan amount.
If your current lender is willing to do the loan w/o a full appraisal, then I have to presume you are working with a portfolio lender (meaning your new loan will NOT be Fannie Mae/Freddie Mac). No way you're going to get a Fannie/Freddie loan w/o a full appraisal - with one exception and that couldn't apply in your situation.
And what that entails is you will probably pay a higher rate w/o Fannie/Freddie backing. The lender may be saying the appraisal is no charge, but they have that price all wrapped up in your rate. If someone is driving by your house, you will pay for it in the end - unless you cancel the transaction.
You also did not mention how long you have owned the property. For Fannie/Freddie, you cannot do a cashout refinance and use the appraisal value until you have owned the home for at least 12 months.
It matters a heck of a lot whether the loan will be primary or for an investment property. Its apples and oranges. If you want to refi a primary, the rate COULD be much lower depending on everything else on your app.
Deciding that a home is or will be your primary residence can be a nightmare these days because underwriters tend to disbelieve the borrower. Borrowers have been caught being dishonest about this countless times. Be prepared for a whole new can of worms if you intend to "say" it will be your primary while you still own and live in another property.
Sorry I can't be more helpful about the rate. I can only say that for an investment property, if you have great credit (over 720), at least 25% equity (AFTER) the cashout, a loan amount if at least100k, with taxes/ins escrowed, and a non-portfolio loan, the rate could be lower than 6.5% as of Friday, 2/18.
But sounds to me like you are working with a private lender and they might be giving you "their" deal. That makes it more difficult to compare from one bank to the next. Get the Good Faith Estimate and look at the fees. That's a start.
Realize that if you have not owned the home for 12 months, do not have a high credit score, and do not want to have a complete appraisal, you will pay a higher rate for that portfolio loan. That bank's process might get you into the loan quicker and easier, but it will not be cheaper.
Lastly, new Fannie/Freddie/FHA loans require the borrower to give a copy of the complete tax return for the prior year. A portfolio lender might not require it. Oftentimes, people have entries on a tax return that kill the mortgage approval.
Nothing easy about mortgages in this environment.
Good luck.