Tickets required for St Peter's Basilica?

No
The Basilica is free

Oh, St Peter’s is so amazing! Enjoy! The first time we were there, we climbed the stairs to the cupola which was a bit claustrophobic but worth it! If available, I highly recommend!

Line up before 7am before it opens or go later around 6pm before it closes.
It is too bad your only option is to go on one of the busier days but the trick is don't show up between 8 am and 4 pm to get in otherwise you will spend a long time in a long line.

The other option is to buy a tour ticket to the Vatican Museum that includes entry into the Basilica from the inside.

The first time we were there, we climbed the stairs to the cupola
which was a bit claustrophobic but worth it!

To mention, while the basilica is free, the dome is not. You would need to purchase tickets at the desk inside the church (yes, do get into the security line for the basilica at 7:00 or before!) for that part. From the website (translated from Italian):

Summer hours: April 1st - September 30th: open at 7.30 am, closes at 5:00 (17:00) PM
- Lift to the terrace level and continue on foot (320 steps) Ticket € 10.00
- Ascent 551 steps on foot, Ticket € 8.00

https://www.vatican.va/various/basiliche/san_pietro/it/cupola/orari.htm

The other option is to buy a tour ticket to the Vatican Museum that
includes entry into the Basilica from the inside.

Jazz, that passage has been closed to everyone, even tours, since June of 2021. No idea if or when it will re-open.

You do have to go through a security line. Please remember it is an active, working place of worship (especially on a Sunday) and you're not supposed to get in the way of any services or worshippers.