I'm close to being done building up a frame and just installed the FD with new shift cable.
I currently have an Ultegra 6503 FD and 105 triple brifters.

Now when I try to get the bigger chainrings, I get two clicks to move it to the middle chainring. When I shift some more, I can only get it to click only once more and is not enough to get it to the big chainring. I've been playing with the cables and still no luck.
I've also checked the outer limit screws and it's not it.

All of the drivetrain components are coming from another frame. The chainstay length are a bit different. I'm coming from a 415mm -->418mm. is that a significant difference?

Also I noticed that when it did go to the big chainring and pedaled with my hands, I heard some rubbing noises coming from the rear wheel/RD. I don't hear it when the chain is in the granny or middle ring.
Is this also stemming from the chain length?

Thanks!

Is the chain still on the granny gear on back when you try this.
You're not supposed to cross chain like that.
How many speeds is the cassette?

It's a 9-speed.
When I attempt to shift to the big chainring, it's on the middle or high cog.
I try to avoid cross chaining.

Check this out.
https://www.parktool.com/repair/readhowto.asp?id=75

Try putting it onto the big ring by pulling on the cable ie not using your levers. If it goes on this way it indicates that it is the cable adjustment.
Cable adjustment is performed by having it set up on the middle chainring and largest rear cog and then adjusting cable so that the chain is as close as possible to the inner cage plate without actually touching

start at the beginning

The derailleur should be a flat screwdriver head above the big chain ring 1/16". The outside of the cage runs also even with the outside chain ring.

Next DON'T HOOK UP THE CABLE YET! Put your rear derailleur in 1st or the lowest gear. Turn the crank and adjust the "L" so it almost rubs the the chain. Depending how much flex your crank and B/B will have will make the chain rub. This has nothing to do with the adjustment.

Now Click you shiffter all the way to low.

Pull the cable taunt so when you twing it. it moves like a loose gituar string.

The second click is called TRIM which is designed to move the derailleur over just a bit so the chain won't make a groove in you cage.

Here is where you will get stuck... It is OK to bend the derailleur cage!!!. Taking what I use is a cresent wrench. on the INSIDE of the cage. Bend a very small "J" pointing the tip to the larger sprocket, This should do the Job.

Now shift the rear derailleur to the smallest. Click the shifter one more time and adjust the "H" so the outside of the cage almost touches...

May the force be with you