Should I put down a lower down payment?

You may want to consider putting down a lesser amount for your down payment thus maximizing the loan amount you are able to obtain based on your specific scenario. This may be helpful later on during your project in the case of cost increases during construction.

Often times there are cost overages to your budget during the construction phase. Unless already built into your loan you will be required to pay for those costs. If you can hold on to as much of your cash as possible, you can be prepared for any expected or unexpected cost increases. You can always pay down your loan at the end of the project before the permanent phase starts in order to lower your monthly payment.