How to adjust a scope for longer ranges than it's main zero?

I finally decided on a scope for my AR build. I went for a Nikon Monarch 3 4-16 BDC. Now I will plan to zero it at 100 yards but questions about compensating at longer distances. I don't really use scopes save for my hunting rifle which is zeroed at 200 yards and it's.270win so it shoots flat already. Everything else is irons and EOTECH and red dots and I know those work in regard for compensating without having to adjust the elevation knobs.

I downloaded the nikon BDC app and it say for example " Set sights to shoot -3.04 inches low at 200 yards for a 100 yard zero" and "set sight to shoot -12.29 inches low at 300 yards for a 100 yard zero" and so on.

Does that mean if the scope is zeroed at 100 yards and the scope is made so each click is 1/4" at 100 yards would I change the elevation by 12 and 25 clicks UP respectively since the numbers are negative to have the scope zeroed in at the cross hairs at those ranges? Or am I misunderstanding those statements?

Added (1). If I'm correct in what I think it is telling me just say it plainly.

Added (2). I should of put 49 clicks for 300 yards

You're more or less correct, in that you're going to have to dial "UP" on the scope to zero at longer ranges, if you believe the numbers. 49 clicks would be a drop of a foot from 100 to 300 yards, though, so your math is probably off someplace. Of course, your rifle and the load you shoot in it won't be what they say, so you won't really know until you've shot a lot at those ranges.

There's a reason I dislike BDC scopes, and you point out one reason, you have all this information in front of your eyes and you're still trying to do math in your head on how to get out to 200 or 300 yards. You did the right thing with the 270 by having a 200 yard ZERO, should do the same for the AR, just set to a 200 ZERO and use the stadia (or circle sets in the Nikon case) for your 300 yard adjustments, then whatever else you can register on the scope points by empirical means--yeah go out and blast away for a while at different ranges and see what lines up.
For a varmint rifle scope I would much rather use a Mil-Dot setup and hold a ZERO without dialing the turrets.
You might have been better served by a 223 specific scope instead of the Monarch 3, but it is a nice scope.
I run my 223 at a 200 yard ZERO just like my 30 calibers and just hold over as needed. I don't have a laser range finder so I'll be guesstimating, but i still like my odds with a clean scope over a BDC.
EDIT: Was just thinking the BDC Nikons SHOULD be set for a 200 yard ZERO, but the 3 might be different.

I have a similar scope (Nikon 4.5-14x BDC) and shot at 200/300 regularly using a 223 Rem bolt gun. Mostly, it was a gun for my wife so she could shoot while I shot my 30 cals. For her, instead of changing the zero (and because I didn't trust Nikon's adjustments), I simply told her what the hold overs were using marks from the BDC. That worked for her on little 6-9 inch steel plates at 300 yards.

She only had to put the target right between the first and second circle at 300, and at the top of the first circle at 200 to be right on with two different loads (52/69 grain bullets).

But short answer for you, when you dial in elevation to compensate for drop, you dial it UP. It actually moves the cross hairs down to life the muzzle up for the same point of aim.

Only thing is, your click values seem off. You should be dialing in around 4 MOA of elevation at 300 yards, which for most.25@100 adjustment values, is 16 clicks or so, not 49.

If its any consolation… I gave up on trying to figure this out with my Golden Eagle.06 and a PRF/ BDC scope back around 1978 or so- when my brain was still functioning. No matter what I did… It was wrong. So it sits in the gun locker… Nearly 40 years now.
What to do? Maybe sell it for an AR?

Edit: yeah, I know… You don't have to blurt it out.

I have found that a web site sponsored by Nikon works for me. Go to "Nikon Spot On" and follow the instructions. It is free and works. I must have 12 Nikon scopes.

1- Learn ballistics, trajectory ballistics this will make sense for what follows
2- sight your rifle to be spot on at 25 yards. First bore sight in at 25, Then the easiest way from a rest and or a gun vise. Fire on shot with cross hires on center of bulls eye, return cross hairs to center and HOLD RIFLE STEADY, adjust cross hires over point of impact. Now you should be right on a 25 yards.
3- Move out to 100 yards and final adjustments. The simplest way for me is to adjust cross hires over point of impact, but the rifle has to be held still.
BTW for the most part a 25 year zero is optimum for point blank range. For muzzle velocities above 3000 fps you may want to sight in a little low at 25 yds.

It means that if you are zeroed for 100 yards, for every click up point of impact is 1/4 inch up.