Anything is possible with the proper planning. You just have more planning to do! ;> Personally, I would say go for it (if it's ok with your husband- you don't want to start any problems) But definitely "doable"!
I drove down from Boston to Florida with my three little ones - at the time 4yrs old and my twins were about 8 months. My biggest concern on the road with them was how long they slept. Normally you would think, yea they're sleeping great, but it was too much. I would think that might be the case with your newborn, since they tend to sleep so much anyway. So, it might take you three days driving instead of two, but try not to go more than three hours and then stop for about an hour. If you can search online for some parks to stop at on the way-something right off the highway- even a McD's with a playground. Just to let them out to stretch and get some energy out. Bring a blanket you can layout at the rest stops picnic areas so the baby can stretch out also! Plan your meal stops, so you don't go past meals. Lunches you can probaly pack in the cooler, bring cereal and milk for breakfast, then you only have to buy dinner. Also plan your hotel stops so you can look around and reserve ahead of time and find somewhere with a good rate instead of the first thing you find when you are tired.
Also bathroom breaks - diaper changes -etc. Have a bag in the car with you with extra clothes on hand, so you don't have to dig through your luggage looking for something. Diapers seemed to leak more since they are sitting so much. Put a plastic bag with a baby blanket on top, under the babies in the car seat, just in case it leaks you won't end up having to wash the whole seat cover (like I did!) LOTS of wipees, diapers, formula, extra bottles & nipples (if your not breastfeeding), water, snacks, small cooler
Also a bag with things for your 3 yr old & 16 mo old to do. My 4 yr old was very good, and was either sleeping or looking out the window most of the time. It might help to have some audio books for kids, you can get them at the library. maybe dolls? I don't know if I would do coloring books, because the crayons will fall and riot may start! anything else they like to do, or just let them watch out the window. Don't offer unless they ask, or you will be on an endless search for something to entertain them with.
I got my son a map so he could see how far we were going and put the different states in different colors so he could follow along (sort of). You may be able to do similar or else just a map so they can feel included and forget the geography! Either way, be sure to show a map before you leave of how far it is from your house to somewhere she's been and then compare to house to your grandparents house- so they can get the idea of how far the trip is. Break the trip into days. Plan the trip with her and make your 3yr old your co-pilot, so she is excited about it- thereby avoiding the are we there yets! Be excited about the trip also, don't show her you are worried that she might be bored or it be a problem at all. Kids always find something to entertain themself with. Special treat at the end of the day is fun to look forward to.
Also, if you are able, try to drive at night, since they will be sleeping anyway. (Personally, this is not a safe option for me, because I am not a good night driver- but for my husband not a problem- you know yourself best) You could get up extra early instead and get some travel time in before breakfast.
Good luck and I hope either way you decide it works out for the best for you and your family.
May you and your family be blessed.
A.