"I am referencing this first screen:
https://gtr1.net/2013/?~Liquid_consolidators::nas1...

I am able to see current picks if I click "RUN SCREENER" at the bottom (e.g., PLTR, ZS, APP, etc...)"

You have to subscribe to AAII's SIPro so they don't complain when the gtr tester gives you current picks (it won't give past ones). Robbie periodically updates the password the gtr tester requires you copy and paste or type in it to get current picks. But I am not keeping track of when he updates his data i.e. there is always the risk he hasn't as of whenever you are seeking the current picks. So you can just go look them up yourself since the screen is not using complicated fields. Various online sites let you get monthly and 200 day moving averages. You can use googlesheets. The Nasdaq100 list is online in various places. barchart.com - Technical analysis is an example of a site that lets you get moving averages. Finviz has the Nasdaq100 list. https://www.slickcharts.com/nasdaq100
Yahoo Finance: NASDAQ-100 (^NDX) Components
also

" Is there a way to replicate this screen/results using SI Pro or something else?"

You can't replicate the backtest anywhere else as far as I know, at least easily. You can get current picks as stated above.
"
Is there any way to use that link above (the screen link I just posted) to identify exactly what is in the portfolio at each date?"

No.

"this is the historic CAGR/IRR (27.4%)"

Yes that is the compound annual growth rate. You can stick in whatever spread rate you prefer as most realistic.