We have been to Bali twice in the past 2 1/2 years as well as 12-14 years ago. We have alwats had a guide so I will give you my impressions based upon observations. You really have to be a very careful but also courageous driver to go it alon in and around Ubud, Sanur, etc. There seems to be a fair amount of "bump and run if an accident does occur. I consider myself a challened driver with our Jag XKR and BMW M3 series Neither one of those cars has less than 400HP and hugs the road and I would not drive their if thast is any criteria. I guide is $50-$70 for a 10 hour day. We used a fellow Putu who was a careful driver who charged $65 for 10 hours thou we were tired after8 or 9 hours. He is knowkedge but a little on the quiet side and once we started asking questions he would offer us full explainations better than many guides we have hadelsewhere. I would just urge you not to use his cousin because thoiugh he is an excellent driver he seemed to have klimited knowledge the one day Putu could not make it.
I would think you would use a guide for 4-5 days of a 19 day stay and you could likely cut a deal for $300 for 5 days.

A driver can be as flexible as you'd like. You don't have to plan out a whole day with your driver. If you are out touring and your child is fussy, you can go back to your hotel.

Driving yourself is a bit risky. Roads are not well-marked, and you won't be accustomed to the local mores in terms of driving. If you are in any kind of accident, it will automatically be your fault. Having a driver is really no more expensive than renting a self-drive car.

i drove the last time i was in bali...it is not hard you just need to pay extra attention as the bali drivers are erratic and lots of people, cars and animals dart into and out from the road.. i decided i would not do it another time...

driving up north is very easy.. if you drive i would encourage you to this leg too.

honestly a driver is so cheap why not just have one the whole time and let the baby nap whenever.. car rental might even cost more

Long-time expats are divided on the subject -- many drive themselves fearlessly and aggressively. Many wouldn't be bothered and leave driving to the locals. I'm with the latter camp.

As noted, drivers are cheap and flexible. Bali drivers -- including taxi drivers -- don't mind waiting, even for hours. This is part of the job and is quite economical.

You can even take a driver on an overnight trip. He'll tell you how much he needs for meals and accommodation and you won't have to worry about him.

I don't advise self driving. The rules of the road (such as they are) are inexplicable to anyone who is not Indonesian. There are all kinds of obstacles and distractions -- double parked cars, potholes, processions, dogs sleeping in the middle of the road, and especially motorcyclists both locals and tourists who don't have a clue.

It's not relaxing for the driver, and an accident can have serious consequences for the foreigner. Don't expect the police to be there for you. (Or be there at all except for an opportunity for money to change hands.)

Usually people who live in Bali have regular drivers whom they call when they need transportation. In the south taxis are plentiful and you can "retain" one for a day or half a day, on or off the meter.

I think taxis are less common in Ubud, but individual cars are plentiful. You should ask the owners of the villa what they do for transport. The staff will probably be able to recommend someone.

10 hour car and driver maybe $50 or less

Do yourself a favor: hire a driver. It's so inexpensive and so convenient.
See if you can get one with a van.
That way you may be able to put your child in the third row and you can have the space in the second . Driving in Bali is positively crazy.... been there several times and would never drive myself.