If you are trying to keep options open, I would not put down a larger down payment than is needed. It's not worth tying up the substantial extra amount needed in the non-liquid house just to lower your monthly payments by a few hundred dollars.
As others have pointed out, if you want to improve cash flow, you should definitely consider a 30 year mortgage. It will give you much more flexibility. You can always choose to pay it off in a shorter timeframe if that makes sense for your situation.
Personally, I would do a 30 year mortgage with 20% down, don't touch your Roth or brokerage account, and invest additional house payoff money. Then down the line you can lump sum the house payoff if you want, or keep that money invested if it makes sense.
How carefully have you explored increased expenses related to your soon to be in the world child? There will be a lot of extra expenses:
- Diapers & wipes
- Food
- Clothes
- Bottles, nipples, pacifiers
- Crib
- Playpen, bouncy chair, swing, etc
- Books, toys
- Childcare
Daycare is by far the biggest expense unless you have a family situation that you can consistently rely on to look after your little one.