So I can definitely readjust later? Or do I need to go back and renegotiate the pre-approval letter?
I think what the other poster meant was if you talk with your bank and point out that you'll be putting down 20% rather than 5%, then they should give you a lower rate than 5.25%. Putting down 20% will also (or it used to anyway) get you out of PMI, which will lower your overall loan costs quite a bit (or it used to anyway).
I started with my own bank, since I thought they would be most favorable to my situation (they have my records and easiest proof of self-employment income).
While your own bank might be the easiest, they are very unlikely to give you the best offer. As an example, I like using USAA for insurance, but they've never been competitive on their 30-year rates. What
@sol said is right - you need to shop around and you also need to educate yourself on origination fees and discount points.
Another thing you should probably look at is a 10 year fixed or even a 5 year fixed if you can find one. Since you can afford the higher payment, shortening the loan term is one of the easiest ways to get a lower rate. There are other products too, like a 3/1 ARM or 5/1 ARM, which might give you lower rates too.