You have to prevent new sod from drying out without watering so often that it lives in constantly wet conditions. That's maybe 3 or 4 times a day for the first week or two. Then start tapering down. You just have to check the soil under/around the sod and plugs to know if it needs water. If it dries, it's finished. I would advise reading some of the google results about sod care.

I would instead just look at getting the fake grass setup going and not have to worry about keeping real grass up and running. How large of an area are we talking?

How big a dog? I bought 9 of these for our 15 pound dog. We use them in the bathroom, but you could use them on your balcony. When I get down to 1 or two, I throw them in the washer using hot water.

https://www.bedbathandbeyond.com/store/product/signature-21-x-34-bath-mat/3350559?skuId=63236057&poc=244435

ETA: I bought white so I could see if they are clean and bleach if needed, but I haven't needed to bleach.

Deep breaths. :-) Yeah, this whole thing is a pain in the butt, isn't it? We're both stuck at home--a diabetic and me, with atrial fib and atrial arrhythmia. Either of us gets this and it's problematic. And I have to go for an ultrasound next Friday--I'm wearing trash clothes I paint in, a P-95 mask, gloves, and my lab coat I use for lye mixing, then throwing everything in the washer when I get home and heading straight for the shower. Wish me luck. We're shopping once a month when we need our meds and otherwise living off canned goods. I've learned an appreciation for canned spinach.

I'm sorry to hear about your child, and that you lost your job. I'm out of work at the moment (they're going to take me back as soon as the county reopens, probably in June, though!)

First things first. As noted, sod must stay damp, and works best if planted in soil, so browning would be expected. Keep it damp at all times.

Urine is going to be way too strong for the sod--it's effectively a really, really strong nitrogen fertilizer. Right after Fido pees, you're going to want to flush the urine out of the sod with at least four times as much water as the volume of the pee--literally until it runs out of the sod and flows away.

I don't know how realistic that is on your deck, as it might lead to it running off and hitting somebody else's deck. What you're trying to do is get it below the root systems of the sod until the sod has time to root into the soil (in normal soils).

Otherwise, burning the sod would be the normal result.

Tigereye's idea is a good one, or fake grass might also work if your dog will adjust to it. That, at least, won't burn and won't confuse the dog into thinking it's something from indoors.

You might also simply try an old waste towel from inside and washing it as necessary in the sink if the dog will adjust to it.